FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
ITAY NEEMAN
Abstract. We present an approach to forcing with finite sequences of models that uses models of two types. This approach builds on earlier work of
Friedman and Mitchell on forcing to add clubs in cardinals larger than ℵ1 ,
with finite conditions. We use the two-type approach to give a new proof of
the consistency of the proper forcing axiom. The new proof uses a finite support forcing, as opposed to the countable support iteration in the standard
proof. The distinction is important since a proof using finite supports is more
amenable to generalizations to cardinals greater than ℵ1 .
MSC-2010: 03E35.
Keywords: forcing, finite conditions, proper forcing axiom.
1. Introduction
There is by now a long history, tracing back to the work of Todorčević [11], of
using finite increasing sequences of countable models as side conditions in forcing
notions, to ensure properness of the resulting poset, and in particular ensure that
ℵ1 is not collapsed.
More recently Friedman [2] and Mitchell [6] independently discovered forcing
notions that add clubs in θ > ℵ1 using finite conditions, while preserving both ℵ1
and θ. The Friedman and Mitchell posets use countable models as side conditions
to ensure preservation of the two cardinals. The side conditions are no longer
increasing sequences; rather they are sets, with various agreement and coherence
conditions on the models in them.
In this paper we reformulate this approach using models of two types, countable
and transitive, rather than just countable. This allows us to return to a situation
where side conditions are increasing sequences, simplifying the definition of the
poset of side conditions.
We show how the resulting poset can be used for the initial Friedman and Mitchell
applications, for an additional application which involves collapsing cardinals in
contexts where it is important not to add branches to certain trees in V , and most
importantly for a new proof of the consistency of the proper forcing axiom.
The original proof of the consistency of PFA used preservation of properness under countable support iterations. The use of countable support makes it impossible
to apply similar ideas for forcing axioms that involve meeting more than ℵ1 dense
sets (in posets that admit master conditions for more than countable structures).
The proof we give uses finite support, and instead of appealing to preservation of
properness, which fails for finite support iterations, it incorporates the two-type
side conditions into the iteration, using them to ensure preservation of ℵ1 , and of
a supercompact cardinal that becomes ℵ2 . This finite support proof has analogues
This material is based upon work supported by the National Science Foundation under Grants
No. DMS-0556223 and DMS-1101204.
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that yield forcing axioms for meeting more than ℵ1 dense sets, but these will be
handled in a separate paper.
The poset of sequences of models of two types is presented in Section 2. We
present it in greater generality than we need. For most applications we only need
the following finite version. Let K be a structure satisfying a large enough fragment
of ZFC. Let S be a collection of countable M ≺K with M ∈ K. Let T be a collection
of transitive W ≺ K with W ∈ K. Suppose that M ∩ W ∈ W and M ∩ W ∈ S
whenever M ∈ S, W ∈ T , and W ∈ M . (This can be arranged in many different
settings, for example if all elements of T are countably closed, and S consists of all
countable elementary substructures of K.) Conditions in the finite two-type model
sequence poset associated to S and T are simply ∈-increasing sequences of models
from S ∪ T , closed under intersections. More precisely a condition is a sequence s
of models M0 ∈ M1 ∈ . . . Mn−1 , where Mi ∈ S ∪ T for each i < n, and so that for
every i, j < n, the intersection Mi ∩ Mj appears in the sequence. Conditions are
ordered by reverse inclusion.
For Q a model that appears in s, the residue of s in Q, denoted resQ (s), is
the subsequence of s consisting of models of s that belong to Q. We prove that
the residue is itself a condition (meaning that it is ∈-increasing and closed under
intersections). We also prove that if t ∈ Q is a condition that extends resQ (s), then
s and t are compatible. This is the most important result proved in Section 2. It
allows deducing that the poset of two-type model sequences is strongly proper, in
a sense defined by Mitchell [6].
The basics of strong properness are presented in Section 3, and are connected
to the poset of two-type model sequences in Section 4. Then in Section 5 we
present the initial applications, yielding the models of Friedman and Mitchell, and
the method for collapsing cardinals to κ+ without adding branches of length κ+
through trees in V .
Finally in Section 6 we use the two-type model sequences to prove the consistency
of PFA with finite supports.
The main definitions and results in this paper, including the finite two-type
model sequences and their use for a finite support proof of the consistency of PFA,
were presented in Neeman [9], which also goes further and indicates how forcing
with finite conditions helps in obtaining higher analogues of the proper forcing
axiom. Since then there have already been some applications of the two-type model
sequences, for example by Veličković–Venturi [13], using side conditions to obtain
new proofs of results of Koszmider, adding a chain of length ω2 in (ω1ω1 , <Fin ),
and of Baumgartner–Shelah, adding a thin very tall superatomic Boolean algebra.
Earlier applications of the Friedman and Mitchell side conditions include Friedman
[3], showing that PFA does not imply that a model correct about ℵ2 must contain
all reals, and Mitchell [7], showing that I(ω2 ) can be trivial.
2. The model sequence poset
Fix cardinals κ < λ. Typically in uses later on λ will be the successor of κ.
Most often in fact we will take κ = ω and λ = ω1 . Fix a transitive set K so that
κ, λ ∈ K, and (K; ∈) satisfies some large enough fragment of ZFC. More precisely we
need enough of ZFC in (K; ∈) to imply that K is closed under the pairing, union,
intersection, set difference, cartesian product, and transitive closure operations,
closed under the range and restriction operations on functions, and that for each
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
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x ∈ K, the closure of x under intersections belongs to K, there is a bijection from
an ordinal onto x in K, and there is a sequence in K consisting of the members of
x arranged in non-decreasing von Neumann rank. Typically in applications we will
take K = H(θ) for a regular cardinal θ; these closure properties then hold for K.
Remark 2.1. Our main applications of the results in this section will all be with
κ = ω and λ = ω1 . Many of the definitions and claims in this section are simpler in
this case. We will point out some of the simplifications, mainly to the definitions, in
remarks throughout the section. Most of the simplifications in case κ = ω are due
to the fact that all models are closed under finite sequences. Similar simplifications
hold for κ > ω if we restrict to models that are <κ closed.
Definition 2.2. S and T are appropriate for κ, λ, and K if:
(1) T is a collection of transitive W ≺ K, and S is a collection of M ≺ K with
κ ⊆ M and |M | < λ. All elements of S ∪ T belong to K, and contain
{κ, λ}.
(2) If M1 , M2 ∈ S and M1 ∈ M2 , then M1 ⊆ M2 .
(3) If W ∈ T , M ∈ S, and W ∈ M , then M ∩ W ∈ W and M ∩ W ∈ S.
(4) Each W ∈ T is closed under sequences of length < κ in K.
Remark 2.3. In case κ = ω and λ = ω1 , condition (1) simplifies to requiring
that elements of T and S are, respectively, transitive and countable elementary
submodels of K that belong to K. Moreover conditions (2) and (4) can be dropped
in this case, as they follow from the elementarity in condition (1).
Suppose that S and T are appropriate. We define a poset P = Pκ,S,T ,K that we
call the poset of two-type model sequences, or simply the sequence poset, associated
to κ, S, T , and K. Conditions are sequences of models in S ∪ T , satisfying certain
requirements that we specify in the next definition.
Definition 2.4. A condition in P is a sequence ⟨Mξ | ξ < γ⟩ of length γ < κ, that
belongs to K, and so that:
(1) For each ξ, Mξ is either an element of T or an element of S.
(2) The sequence is increasing in the following sense: for each ζ < γ, the set
{ξ < ζ | Mξ ∈ Mζ } is cofinal in ζ. In particular for successor ordinals
ζ < γ, Mζ−1 ∈ Mζ .
(3) For each ζ < γ, the sequence ⟨Mξ | ξ < ζ ∧ Mξ ∈ Mζ ⟩ belongs to Mζ .
(4) The sequence is closed under intersections, meaning that for all ζ, ξ < γ,
Mζ ∩ Mξ is on the sequence.
Note that condition (3) holds automatically for transitive nodes Mζ . It follows
for such nodes from the closure of Mζ in K, the fact that the entire sequence ⟨Mξ |
ξ < γ⟩ belongs to K, and enough closure for K to produce ⟨Mξ | ξ < ζ ∧ Mξ ∈ Mζ ⟩
from Mζ and ⟨Mξ | ξ < γ⟩.
Remark 2.5. In case κ = ω and λ = ω1 , condition (3) of Definition 2.4 and the
requirement of membership in K hold automatically, and condition (2) requires
simply that Mζ−1 ∈ Mζ for each ζ > 0. Thus the definition in this case simplifies
to the following: ⟨Mξ | ξ < γ⟩ is a finite ∈-increasing sequence of elements of S ∪ T ,
and is closed under intersections.
Claim 2.6. If ξ < ζ, then Mξ has smaller von Neumann rank than Mζ .
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Proof. Immediate by induction on ζ using condition (2).
Abusing notation slightly we often refer to a condition as a set {Mξ | ξ < γ}
rather than a sequence. There is no loss of information in talking about the set
rather than the sequence, since by Claim 2.6 the sequence order is determined
uniquely from the elements of the sequence.
Definition 2.7. Conditions in P are ordered by reverse inclusion (with conditions
viewed as sets). In other words, {Mξ | ξ < γ} ≤ {Nξ | ξ < δ} iff {Mξ | ξ < γ} ⊇
{Nξ | ξ < δ}.
Claim 2.8. Suppose that κ is regular, let τ ≤ κ be regular, and suppose that K
and all models in S are <τ closed in V . Then P is <τ closed.
Proof. If κ is regular and K and all models in S are <τ closed, then the union
of a decreasing set of fewer than τ conditions in P is itself, when ordered by von
Neumann rank, a condition in P. This is immediate from the definitions. The
closure is needed for condition (3) in Definition 2.4.
We refer to elements of T ∪ S as nodes. By condition (1) in Definition 2.2, S
and T are disjoint, so each node belongs to exactly one of them. Elements of T are
transitive nodes, also called nodes of transitive type. Elements of S are small nodes,
or nodes of small type. We say that M is of the same or smaller type than N if the
two nodes are of the same type, or M is of small type and N is of transitive type.
Given a condition s = {Mξ | ξ < γ} and nodes M = Mξ and N = Mζ which belong
to s, we use interval notation in the natural way, for example (M, N ) is the interval
of nodes strictly between M and N , namely the interval of nodes Mι , ξ < ι < ζ.
Definition 2.9. s is a precondition if it satisfies conditions (1) and (2) in Definition
2.4. s is a nice precondition if it also satisfies condition (3).
In case κ = ω and λ = ω1 , a precondition is a finite ∈-increasing sequence
from S ∪ T , and niceness holds automatically by closure of all models under finite
sequences.
Claim 2.10. Let s be a precondition.
(1) If M and Q are nodes in s, with M ∈ Q and M of the same or smaller
type than Q, then M ⊆ Q.
(2) If W is a node in s of transitive type, and M is a node in s occurring below
W , then M ∈ W and hence by (1) also M ⊆ W .
(3) If Q ∈ s is of small type, M ∈ s occurs before Q, and there are no nodes of
transitive type in s between M and Q, then M ∈ Q. If in addition M is of
small type then by (1) also M ⊆ Q.
Proof. The first condition is clear, by conditions (1) and (2) in Definition 2.2.
For condition (2), let W ∈ s be of transitive type, and let M occur below W .
By condition (2) of Definition 2.4, there are nodes of s at or above M which belong
to W . Let M ∗ be the least one. By condition (1) of the claim, M ∗ ⊆ W . By
minimality of M ∗ and condition (2) of Definition 2.4 it follows that M ∗ must be
equal to M , and hence M = M ∗ ∈ W .
Fix finally Q and M as in condition (3). Again there are nodes of s at or above
M which belong to Q. Let M ∗ be the first one. If M ∗ is M then M ∈ Q, and
we are done, so suppose M ∗ occurs above M . In this case by assumption of the
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
5
condition, M ∗ is of small type, and hence by condition (1), M ∗ ⊆ Q. As in the
previous paragraph, the minimality of M ∗ now implies that M ∗ is M .
Claim 2.11. Let s be a precondition. Let M1 and M2 be nodes of s with M1
occurring before M2 . Suppose that there are nodes of transitive type between M1
and M2 . Then there are nodes of transitive type between them, that belong to M2 .
Proof. Work by induction on M2 . Let W be a node of transitive type between M1
and M2 . Using condition (2) of Definition 2.4, there is W ∗ occurring at or above
W , that belongs to M2 . If W ∗ is of transitive type, we are done. Otherwise by
Claim 2.10, W ∗ ⊆ M2 . By induction there is a node of transitive type between M1
and W ∗ , that belongs to W ∗ . Since W ∗ ⊆ M2 , the node belongs to M2 .
Claim 2.12. Let s be a precondition. Suppose that s satisfies the following weak
closure under intersections:
(w4) If W and M are nodes in s of transitive and small type respectively, and
W ∈ M (in particular W occurs before M ), then M ∩ W is a node in s.
Then s is closed under intersections, in other words it satisfies condition (4) in
Definition 2.4.
Proof. Suppose s is not closed under intersections, and let Q and M witness this,
with Q occurring before M , and M minimal.
If M is of transitive type, then by Claim 2.10, Q ⊆ M , hence M ∩ Q = Q is a
node of s. So we may assume that M is of small type.
If there are no nodes of transitive type between Q and M , then by Claim 2.10,
Q ∈ M . If Q is of small type then by the same claim Q ⊆ M so M ∩ Q = Q is a
node of s. If Q of transitive type then by condition (w4) M ∩ Q is a node of s.
So we may assume that there are nodes of transitive type between Q and M . By
Claim 2.11 there is such a node W with W ∈ M . By condition (w4) then M ∩ W
is a node of s. It must occur before W , and therefore before M . By induction then
(M ∩ W ) ∩ Q is a node of s. Since W is a node of transitive type above Q, we have
Q ⊆ W by Claim 2.10. Hence M ∩ Q = (M ∩ W ) ∩ Q, so M ∩ Q is a node of s. Definition 2.13. Let s be a precondition and let Q be a node of s. The residue
of s in Q, denoted resQ (s), is the set {M ∈ s | M belongs to Q}.
Claim 2.14. If s is a nice precondition then resQ (s) belongs to Q.
Proof. Immediate from the definition of a nice precondition.
Claim 2.15. Let s be a condition. Let Q ∈ s be of small type, and let W ∈ resQ (s)
be of transitive type. Then there are no nodes of resQ (s) in the interval [Q ∩ W, W )
of s. (Q ∩ W belongs to s by closure of s under intersections.)
Proof. Let N be a node of resQ (s) that occurs before W in s. Then since W is of
transitive type, N ∈ W by Claim 2.10. Since N ∈ resQ (s), N ∈ Q by definition of
the residue. So N ∈ Q ∩ W , and this implies that N can only occur before Q ∩ W
in s.
Definition 2.16. Let s be a condition and let Q ∈ s be of small type. Let W be
a node of transitive type in resQ (s). Then the interval [Q ∩ W, W ) of s is called a
residue gap of s in Q.
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Claim 2.17. Let s be a condition and let Q be a node of s. If Q is of transitive
type, then resQ (s) consists of all nodes of s that occur before Q. If Q is of small
type then resQ (s) consists of all nodes of s that occur before Q and do not belong
to residue gaps of s in Q.
Proof. If Q is of transitive type then by Claim 2.10 all nodes of s before Q belong
to Q, and are therefore nodes of the residue. Nodes from Q upward have equal or
higher von Neumann ranks than Q, and therefore cannot belong to Q, so they are
not nodes of the residue.
Suppose Q is of small type. Again nodes occurring from Q upward cannot belong
to Q because they have the same or higher von Neumann rank, so they are not nodes
of the residue. Nodes below Q that belong to residue gaps are not in the residue
by Claim 2.15. It remains to prove that all nodes below Q that are outside residue
gaps, belong to the residue.
Let N be a node of s below Q, outside all residue gaps. If there are no transitive
nodes between N and Q, then by Claim 2.10, N belongs to Q and therefore N
belongs to the residue. Suppose that there are transitive nodes between N and Q.
By Claim 2.11 there are such nodes that belong to Q. Let W be the first one. Then
[Q ∩ W, W ) is a residue gap. Since N occurs below W and outside all residue gaps,
it must occur below Q ∩ W . By minimality of W there are no transitive nodes
between N and Q ∩ W that belong to Q ∩ W , and hence by Claim 2.11 there are
outright no transitive nodes between N and Q∩W . By Claim 2.10 then N ∈ Q∩W ,
hence in particular N ∈ Q and therefore N belongs to the residue.
Lemma 2.18. Let s be a condition, and let Q be a node of s. Then resQ (s) is a
condition.
Proof. If Q is of transitive type then resQ (s) is an initial segment of s, and is easily
seen to be a condition. Suppose then that Q is of small type. We prove that the
residue satisfies the conditions in Definition 2.4.
The residue belongs to K since it can be obtained from Q and s using simple
set operations. Condition (1) of Definition 2.4 for the residue is immediate from
the same condition for s. Condition (4) for the residue is again immediate, from
the same condition for s and the elementarity of Q in K (which implies that the
intersection of two nodes that belong to Q is itself an element of Q). Conditions
(2) and (3) are clear if Mζ ∈ resQ (s) is of small type, since in this case Mζ ⊆ Q
and any node of s that belongs to Mζ is also a node of the residue. Finally, if
Mζ ∈ resQ (s) is of transitive type, then all nodes of s (and of resQ (s)) that occur
before Mζ are elements of Mζ , by Claim 2.10. Condition (2) for the residue at ζ
follows immediately from this. Condition (3) holds automatically since Mζ is of
transitive type.
Definition 2.19. Two conditions s and t are compatible in P if there is a condition
r such that r ⊇ s ∪ t. The conditions are directly compatible if r can be taken to be
exactly the closure of s ∪ t under intersections.
Lemma 2.20. Let s be a condition, and let Q ∈ s be a transitive node. Suppose
that t is a condition that belongs to Q and extends resQ (s). Then s∪t is a condition,
and in particular s and t are directly compatible.
Proof. K satisfies enough of ZFC that s ∪ t ∈ K. Condition (1) of Definition 2.4 is
immediate for s ∪ t. Condition (2) too is clear, using the fact that s ∪ t is the same
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
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as s above Q, and the same as t below Q. For condition (4), it is enough by Claim
2.12 to verify (w4) of the claim. Fix nodes W ∈ M of transitive and small type
respectively in s ∪ t. If M occurs below Q then both M and W are nodes of t, and
closure of t under intersections implies that M ∩ W is a node of t hence of s ∪ t.
If M and W both occur at or above Q then both are nodes of s, and closure of s
implies M ∩ W is a node of s hence of s ∪ t. If M occurs above Q and W occurs
below Q, then by Claim 2.10, W ⊆ Q, so M ∩ W = M ∩ Q ∩ W . M ∩ Q is a node
of s below Q, hence a node of resQ (s), hence a node of t. W is also a node of t. By
closure of t under intersections it follows that M ∩ Q ∩ W is a node of t, hence a
node of s ∪ t.
Finally, condition (3) for s ∪ t is clear if Mζ occurs below Q, since the part of s ∪ t
below Q is simply t. The condition is also clear if Mζ = Q, since Q is transitive. If
Mζ occurs above Q, then {N ∈ s ∪ t | N ∈ Mζ } = {N ∈ s | N ∈ Mζ } ∪ {N ∈ t |
N ∈ Mζ ∩ Q}. The left part of the union belongs to Mζ since s is a condition, and
the right part of the union belongs to Mζ (in fact to Mζ ∩ Q) since t is a condition
and Mζ ∩ Q is a node of t. The entire union then belongs to Mζ by elementarity of
Mζ in K.
Lemma 2.21. Let s be a condition and let Q ∈ s be a small node. Suppose that
t is a condition that belongs to Q and extends resQ (s). Then s and t are directly
compatible.
Proof. We first show that s ∪ t is increasing. (The proof of this will use the assumption that s is closed under intersections.) We then add nodes to s ∪ t that close
it under intersections, and show that the sequence resulting from the addition of
these nodes is a condition. Since the sequence we generate is obtained from s and
t using simple set operations (union, and closure under intersections), it belongs to
K. So we will only have to worry about the other clauses in Definition 2.4.
Claim 2.22. s ∪ t is increasing, and is therefore a precondition.
Proof. Let u consist of the nodes of s above Q. It is clear that t ∪ {Q} ∪ u is
increasing, or more precisely that it satisfies condition (2) of Definition 2.4, when
ordered in the natural way, namely the nodes of t ordered as they are in t, followed
by Q, followed by the nodes of u ordered as they are in s. The reason is that t is
a condition and hence increasing, t ∈ Q hence t ⊆ Q by condition (1) in Definition
2.2 and the fact that |t| < κ, and {Q} ∪ u is a tail-end of the condition s and hence
increasing.
Since t extends resQ (s), by Claim 2.17, the only nodes of s that do not belong
to t ∪ {Q} ∪ u are the nodes in residue gaps of s in Q. Recall that residue gaps are
intervals in s of the form [Q ∩ W, W ) where W is a transitive node of s that belongs
to Q. In particular W belongs to t and hence to t ∪ {Q} ∪ u.
We prove that the sequence obtained from t ∪ {Q} ∪ u by adding the nodes of
each residue gap [Q ∩ W, W ), immediately before W and ordered inside the interval
according to their ordering in s, is increasing. Since the resulting sequence has all
nodes of s ∪ t, this establishes the claim.
Since t∪{Q}∪u is increasing, and each residue gap is increasing (being a segment
of a condition), it is enough to check condition (2) of Definition 2.4 at the borders
of each residue gap [Q ∩ W, W ). At the higher border the condition follows from
the same condition for s at W , since the residue gap includes a tail-end of nodes of
s below W . At the lower end the condition follows from the fact that t is increasing
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and contained in Q. Since t is increasing, the set of nodes of t which belong to W
is cofinal below W . Since t is contained in Q, all these nodes belong to Q ∩ W . Remark 2.23. It follows from the proof that for any residue gap [Q ∩ W, W ) of s
in Q, no nodes of t occur in the interval [Q ∩ W, W ) of s ∪ t.
Let W be a transitive node of t which does not belong to s. Suppose there are
transitive nodes of s in the interval (W, Q) of s ∪ t, and let W ∗ be the first one.
Let EW list, in order, the small nodes of s starting from Q ∩ W ∗ , up to but not
including the first transitive node of s above Q ∩ W ∗ .
Note that W ∈ M for each M ∈ EW . This is certainly the case for the first
element of EW , namely Q ∩ W ∗ , since W ∈ W ∗ by Claim 2.10, and W ∈ t ⊆ Q. For
M occurring above Q ∩ W ∗ in EW , W ∈ Q ∩ W ∗ ⊆ M , where the final inclusion
holds by Claim 2.10 as all nodes of s between Q ∩ W ∗ and M are small.
Define FW to be {M ∩ W | M ∈ EW }, with the ordering induced by the ordering
of nodes in EW . Since M is small and W ∈ M for each M ∈ EW , each element
M ∩ W of FW is a small node by Definition 2.2. It is easy to check that FW is
increasing, and indeed M ∩ W ∈ M ′ ∩ W whenever M occurs below M ′ in EW .
The lowest node of FW is (Q ∩ W ∗ ) ∩ W = Q ∩ W , and by Definition 2.2, all nodes
of FW belong to W .
Let W be a transitive node of t which does not belong to s, and suppose that
there are no transitive nodes of s in the interval (W, Q). Let EW list, in order,
the small nodes of s starting from Q, up to but not including the first transitive
node of s above Q if there is one, and all nodes of s starting from Q if there are no
transitive nodes above Q.
Again, W ∈ M for each M ∈ EW , since W ∈ Q ⊆ M , with the final inclusion
using Claim 2.10. Again define FW to be {M ∩ W | M ∈ EW }, with the ordering
induced by the ordering of the nodes in EW . Again by Definition 2.2, all elements
of FW are small nodes that belong to W . The first node of FW is Q ∩ W , and again
FW is increasing.
Let r be obtained from s ∪ t by adding all nodes in FW , for each transitive W
that belongs to t − s, placing the nodes of FW in order, right before W . We will
prove that with this specific placement, r is increasing.
Remark 2.24. Note that every node of r that belongs to Q is a node of t. This is
certainly the case for nodes of s, since t extends resQ (s). The only other nodes of
r which need to be checked are the nodes in FW for transitive W ∈ t − s, but these
do not belong to Q: They have the form M ∩ W with M ∈ EW , which implies that
M is a small node that contains either Q or Q ∩ W ∗ , for W ∗ transitive above W .
Either way M ∩ W contains Q ∩ W , which is impossible when M ∩ W ∈ Q (since
then M ∩ W ∈ Q ∩ W ).
Claim 2.25. r is increasing, and is therefore a precondition.
Proof. Since s∪t is increasing, and each added interval FW is increasing, it is enough
to verify condition (2) of Definition 2.4 at the borders of each added interval FW .
At the upper border, every element of FW belongs to W . For the lower border,
we have to show that cofinally many elements of r below Q ∩ W belong to Q ∩ W .
Since all elements of t below W belong to Q ∩ W (as t ⊆ Q), it is enough to check
that t is cofinal in r below Q ∩ W . To check this, note that the only elements of r
below Q ∩ W which do not belong to t are either in (a) residue gaps [Q ∩ W̄ , W̄ ) of
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
9
s in Q, or (b) added intervals FW̄ . The gap or added interval cannot overlap W ,
since W itself is a node of t. (Residue gaps do not include nodes of t by Remark
2.23.) So in both (a) and (b), if the gap or added interval has nodes below Q ∩ W ,
it must be that W̄ occurs before W . Since W̄ is a node of t, in both cases the gap
or added interval is capped by a node of t below Q ∩ W . It follows that the nodes
of t are cofinal in r below Q ∩ W .
Claim 2.26. r is closed under intersections.
Proof. It is enough to verify condition (w4) of Claim 2.12. Fix then W ∈ M of
transitive and small type respectively, both nodes of r. We prove that M ∩ W is a
node of r.
Since the intervals added to s ∪ t to form r consist only of small nodes, W must
be a node of s ∪ t. Suppose first that it is a node of s.
If W occurs above Q, then M must also occur above Q, hence M is a node of s,
and M ∩ W is a node of s by closure of s under intersections. Suppose then that
W occurs below Q. If M is a node of s then M ∩ W is a node of s hence of r.
If M is a node of one of the intervals added to s ∪ t to form r, then it is of the
form M ′ ∩ W ∗ for some M ′ ∈ s and W ∗ ∈ t. W ∗ is above M , hence above W , so
W ⊆ W ∗ . Then M ∩ W = (M ′ ∩ W ∗ ) ∩ W = M ′ ∩ W ∈ s where membership of
M ′ ∩ W in s follows from the closure of s, as both M ′ and W are nodes of s. The
last remaining possibility (in the case that W is a node of s) is that M is a node
of t. Then M ∈ Q and since M is small it follows that M ⊆ Q, so W ∈ Q. W is
then a node of resQ (s), hence a node of t. By closure of t, M ∩ W is a node of t,
and hence of r.
Consider next the case that W is not a node of s. In other words it is a node of
t − s. If M is a node of t, then M ∩ W is a node of t hence of r. If M is a node
of one of the intervals added to s ∪ t to form r, then it is of the form M ′ ∩ W ′ for
some M ′ ∈ s and W ′ above M and hence above W . In this case M ∩ W = M ′ ∩ W ,
so that membership of M ∩ W in r reduces to membership of M ′ ∩ W in r, for a
node M ′ of s. Thus it is enough to consider the case that M is a node of s. If there
are transitive nodes W ′ of s between W and M , we may replace M with M ∩ W ′ ,
since M ∩ W ′ is also a node of s, and M ∩ W = (M ∩ W ′ ) ∩ W . So, in sum, we may
assume that W is a node of t − s, M is a node of s − t, and there are no transitive
nodes of s between W and M . One can check in this case that M belongs to EW ,
hence M ∩ W belongs to FW and is a node of r by definition.
Since the added intervals FW consist only of nodes that are intersections of nodes
in s with nodes in t, it follows from Claim 2.26 that r is exactly equal to the closure
of s ∪ t under intersections. To complete the proof of the lemma it remains to verify
condition (3) of Definition 2.4 for r. The condition holds automatically at nodes
of transitive type. For a small node N of r that belongs to t, we have N ⊆ Q and
therefore by Remark 2.24, {M ∈ r | M ∈ N } = {M ∈ r | M ∈ N ∧ M ∈ t}.
Condition (3) for r at N therefore follows from the same condition for t. We must
check the condition for the other small nodes of r, namely small nodes that belong
to s − t, and small nodes in the added intervals FW .
Claim 2.27. Let N be a small node of s. Then every node of r that belongs to N
is either a node of s in N , or a node of t in N , or the intersection of a small node
of s in N with a transitive node of t in N .
10
ITAY NEEMAN
Proof. Suppose not. Since the only nodes of r that do not belong to s ∪ t are nodes
in added intervals FW , this means that there is a transitive node W of t and a node
M ∈ FW , equal to M ∗ ∩ W say with M ∗ ∈ EW , so that M = M ∗ ∩ W ∈ N , but
M ∗ and W do not both belong to N .
Suppose for simplicity that there is a transitive node of s in the interval (W, Q)
of s ∪ t, and let W ∗ be the least such. The case that there are no transitive nodes
of s in that interval is similar.
Replacing N with N ∩ W ∗ if needed, we may assume that N occurs before W ∗
in s. N cannot occur at or before M in r, since M ∈ N . And it cannot occur in the
interval (M, W ) of r, since this interval is contained in FW which is disjoint from
s ∪ t. So N must occur above W in r. Thus, both M ∗ and N occur between W
and W ∗ in r, and by minimality of W ∗ it follows that there are no transitive nodes
of s between them.
If N occurs before M ∗ , then by Claim 2.10, N ∈ M ∗ . Then N ∩ W ∈ M ∗ ∩ W by
a simple calculation using the fact that W ∈ M ∗ . On the other hand M ∗ ∩ W ∈ N ,
and this implies that M ∗ ∩W ∈ N ∩W . Altogether then N ∩W ∈ M ∗ ∩W ∈ N ∩W ,
contradiction. A similar argument leads to a contradiction in case N = M ∗ .
So it must be that N occurs above M ∗ . By Claim 2.10 it follows that M ∗ ∈ N ,
and that M ∗ ⊆ N . Since W ∈ M ∗ it follows further that W ∈ N .
Claim 2.28. Let N be a small node of s. Then {M ∈ r | M ∈ N } belongs to N .
Proof. By the previous claim, the closure of r ⊇ s ∪ t under intersections, and the
elementarity of N , {M ∈ r | M ∈ N } consists precisely of the nodes of s that
belong to N , the nodes of t that belong to N , and all intersections of these nodes.
Thus it is enough to prove that both {M ∈ s | M ∈ N } and {M ∈ t | M ∈ N }
belong to N . {M ∈ r | M ∈ N } is the closure of the union of these two sequences
under intersections, and belongs to N by elementarity of N .
That {M ∈ s | M ∈ N } belongs to N is clear by condition (3) of Definition 2.4
for s. We prove that {M ∈ t | M ∈ N } belong to N .
We can assume that there are no transitive nodes of s above N : otherwise letting
W be the first such node, we can replace s, Q, and t by s ∩ W , Q ∩ W , and t ∩ W
respectively. We can also assume that there are no transitive nodes of s above Q:
otherwise letting W be the first such node, we can replace s and N by s ∩ W and
N ∩ W respectively.
We now divide into two cases. If N occurs at or above Q, then (using the
assumptions in the previous paragraph, and Claim 2.10) Q ⊆ N , and since t ⊆ Q
it follows that {M ∈ t | M ∈ N } = t ∈ Q ⊆ N . If N occurs below Q, then (again
using the assumptions in the previous paragraph, and Claim 2.10) N ∈ Q, and
since t ⊇ resQ (s) it follows that N ∈ t, hence {M ∈ t | M ∈ N } ∈ N by condition
(3) of Definition 2.4 for t.
Claim 2.29. Let N be a small node of r that belongs to an added interval FW .
Then {M ∈ r | M ∈ N } belongs to N .
Proof. Let N ∗ ∈ EW be such that N = N ∗ ∩ W . Then by the previous claim
{M ∈ r | M ∈ N ∗ } belongs to N ∗ . It is shown in the paragraphs defining EW that
W ∈ N ∗ . Thus by elementarity of N ∗ , {M ∈ r | M ∈ N ∗ } ∩ W belongs to N ∗ .
By closure of W (condition (4) in Definition 2.2), {M ∈ r | M ∈ N ∗ } ∩ W belongs
also to W . Thus {M ∈ r | M ∈ N } = {M ∈ r | M ∈ N ∗ ∩ W } = {M ∈ r | M ∈
N ∗} ∩ W ∈ N ∗ ∩ W = N .
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
11
The last two claims complete the proof of condition (3) of Definition 2.4 for r,
and with it the proof that r is a condition. Since r ⊇ s ∪ t, and all nodes of r are in
the closure of s ∪ t under intersections, this completes the proof of Lemma 2.21. Remark 2.30. In case κ = ω and λ = ω1 , condition (3) of Definition 2.4 holds
trivially, and therefore Claims 2.27 through 2.29 in the proof of Lemma 2.21 are
not necessary.
Corollary 2.31. Let s be a condition and let Q be a node of s. Suppose t is a
condition that belongs to Q and extends resQ (s). Then:
(1) s and t are directly compatible.
(2) Let r witness that s and t are directly compatible, meaning that r is the
closure of s ∪ t under intersections. Then resQ (r) = t.
(3) The small nodes of r outside Q are all of the form N or N ∩ W where N
is a small node of s and W is a transitive node of t.
Proof. Condition (1) is immediate from Lemma 2.20 if Q is of transitive type, and
from Lemma 2.21 if Q is of small type.
If Q is of transitive type, then Lemma 2.20 shows that r is simply s ∪ t, and since
t extends resQ (s) it is clear that resQ (r) = t.
If Q is of small type, then by Remark 2.24, every node of r that belongs to Q is
a node of t, in other words resQ (r) ⊆ t. Since t ⊆ r and t ⊆ Q the reverse inclusion
holds trivially, completing the proof of condition (2).
Condition (3) is clear if Q is transitive, since then all nodes of r outside Q are
nodes of s. If Q is small, then r consists of s ∪ t together with the added intervals
FW from the proof of Lemma 2.21. Nodes in t belong to Q, and nodes in the added
intervals are of the form N ∩ W , for small N ∈ s and transitive W ∈ t, by the
definition of the added intervals.
Corollary 2.32. Let M ∈ S ∪ T , and let t be a condition that belongs to M . Then
there is a condition r ≤ t with M ∈ r. Moreover r can be taken to be the closure of
t ∪ {M } under intersections.
Proof. Let s = {M }. It is clear that s is a condition, and that resM (s) = ∅. Since
t ≤ ∅, by Corollary 2.31 the conditions s and t are directly compatible. Let r witness
this. Then r ≤ t, M ∈ r, and r is the closure of t ∪ {M } under intersections.
Claim 2.33. Let s and t be conditions, and let W be a transitive node that belongs
to both. Suppose that s and t are directly compatible and let r witness this. Then
resW (r) is the closure of resW (s) ∪ resW (t) under intersections.
Proof. Since resW (s) ∪ resW (t) ⊆ (s ∪ t) ∩ W , the closure of resW (s) ∪ resW (t) under
intersections is contained in r, and contained
in W , hence contained in resW (r).
∩
Every node M in r is ∩
of the form i<k Mi where Mi ∈ s ∪ t. If M belongs to
W then M = M ∩ W = i<k Mi ∩ W . By closure of s and t under intersections,
and since W belongs to both s and t, Mi ∩ W ∈ s ∪ t. Since Mi ∩ W ∈ W or
Mi ∩ W = W , this implies that Mi ∩ W ∈ resW (s) ∪ resW (t) ∪ {W }. ∩Components
Mi ∩ W which are equal to W can be dropped from the intersection i<k Mi ∩ W
without affecting its value. So M belongs to the closure of resW (s) ∪ resW (t) under
intersections.
12
ITAY NEEMAN
We end this section with a slight modification of the poset P = Pκ,S,T ,K , which
we call the decorated version. We prove that this modified version satisfies a parallel
of Corollary 2.31.
Claim 2.34. Let s be a condition and let M and N be nodes of s so that M ∈ N .
Let M ∗ be the successor of M in s. Then M ∗ ⊆ N .
Proof. If N is of transitive type, or there are no nodes of transitive type between
M and N , then M ∗ ⊆ N by Claim 2.10. Suppose that N is of small type and there
are nodes of transitive type between M and N . Let W be the least one. By Claim
2.10, M ∈ W . Since M ∈ N it follows that M ∈ N ∩ W , hence in particular N ∩ W
occurs above M . Since N ∩ W occurs below W , there are no nodes of transitive
type between M and N ∩ W . By Claim 2.10 it follows that M ∗ ⊆ N ∩ W .
Definition 2.35. Define Pdec = Pdec
κ,S,T ,K to be the poset consisting of pairs ⟨s, f ⟩
where:
(1) s ∈ Pκ,S,T ,K .
(2) f is a function on the nodes of s. For every M ∈ s, f (M ) is a set of size
< κ.
(3) If M ∈ s is not the largest node in s, then f (M ) is an element of the
successor of M in s. If M is the largest node of s, then f (M ) ∈ K.
(4) f belongs to K, and for every node N ∈ s, the restriction of f to resN (s)
belongs to N .
Note that condition (4) holds automatically for transitive N ; it follows for such N
using the closure of N given by Definition 2.2.
The ordering on Pdec is the following: ⟨s∗ , f ∗ ⟩ ≤ ⟨s, f ⟩ iff s∗ ≤ s, and f ∗ (M ) ⊇
f (M ) for every M ∈ s.
Remark 2.36. In case κ = ω, condition (3) is equivalent to the requirement that
f (M ) ⊆ M ∗ when M ∗ is the successor of M in s, and f (M ) ⊆ K when M is the
largest node of s. Moreover condition (4) is automatically true in this case: By
Claim 2.34, f (M ) ⊆ N for every M ∈ resN (s), and together with the fact resN (s)
and f (M ) for M ∈ resN (s) are all finite, this automatically gives f resN (s) ∈ N .
Finiteness also automatically gives f ∈ K.
Thus, in case κ = ω, the definition simplifies to the following: s ∈ P, f is a
function defined on the nodes of s, and for each M ∈ s, f (M ) is a finite subset of
the successor of M in s if there is one, and of K if M is the largest node.
Let ⟨s, f ⟩ ∈ Pdec , and let Q be a node of s. Then the residue of ⟨s, f ⟩ in Q,
denoted resQ (s, f ), is defined to be ⟨resQ (s), f resQ (s)⟩.
Claim 2.37. resQ (s, f ) is a condition in Pdec , and belongs to Q.
Proof. Conditions (1) and (2) for resQ (s, f ) are clear from the same conditions for
⟨s, f ⟩. Since resQ (s) ∈ Q, condition (4) for ⟨s, f ⟩, used with N = Q, directly implies
that resQ (s, f ) belongs to Q. Condition (4) transfers from ⟨s, f ⟩ to resQ (s, f ):
f resQ (s) belongs to K since it belongs to Q, and for any small node N ∈ resQ (s),
f resQ (s) resN (resQ (s)) = f resN (s) ∈ N , using the fact that N ⊆ Q.
Finally condition (3) for resQ (s, f ) is immediate from the same condition for s,
in all instances except when the successor of M in s is the bottom node of a residue
gap of s in Q. Let [Q ∩ W, W ) be the gap. By condition (3) for s, f (M ) ∈ Q ∩ W .
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
13
In particular f (M ) ∈ W . Since W is the successor of M in resQ (s) this establishes
the M instance of condition (3) for resQ (s, f ).
Two conditions ⟨s, f ⟩, ⟨t, g⟩ ∈ Pdec are directly compatible in Pdec if there is ⟨r, h⟩
witnessing their compatibility, with all nodes of r obtained by intersections from
nodes of s ∪ t.
Claim 2.38. Let ⟨s, f ⟩ ∈ Pdec , let Q be a node of s, and let ⟨t, g⟩ ∈ Pdec be an
element of Q that extends resQ (s, f ). Then ⟨s, f ⟩ and ⟨t, g⟩ are directly compatible
in Pdec .
Proof. By Corollary 2.31, s and t are directly compatible in P. Let r witness this.
It is enough now to find h so that ⟨r, h⟩ ∈ Pdec , for every M ∈ s, h(M ) ⊇ f (M ),
and for every M ∈ t, h(M ) ⊇ g(M ).
Set h(M ) to be equal to g(M ) for M ∈ t, equal to f (M ) for M ∈ s − t, and equal
to ∅ for M ∈ r − (s ∪ t). Since ⟨t, g⟩ ≤ resQ (s, f ), g(M ) ⊇ f (M ) for M ∈ s ∩ t. It
follows that h(M ) ⊇ f (M ) for M ∈ s. By definition, h(M ) ⊇ g(M ) for M ∈ t. It
remains to check that ⟨r, h⟩ satisfies the conditions in Definition 2.35.
Conditions (1) and (2) are clear. Fix M ∈ r for condition (3). If M is the largest
node of r then M is the largest node of s, and h(M ) = f (M ) ∈ K. If M ∈ s − t
then M is either at or above Q, or in a residue gap [Q∩W, W ) of s in Q. Either way
the successor M ∗ of M in r is the successor of M in s, and h(M ) = f (M ) ∈ M ∗ .
If M ∈ r − (s ∪ t) then h(M ) = ∅, which certainly belongs to M ∗ . It remains to
consider M ∈ t. If Q is of transitive type, then t is an initial segment of r, M ∗ is
either the successor of M in t or M ∗ = Q. h(M ) = g(M ) ∈ M ∗ by condition (3)
for ⟨t, g⟩ in the former case, and since ⟨t, g⟩ ∈ Q in the latter. Suppose Q is of small
type. If the successor M ∗ of M in r is equal to the successor of M in t, or to Q,
then h(M ) ∈ M ∗ as in the case of transitive type Q. If M ∗ is the bottom node of
a residue gap [Q ∩ W, W ) of s in Q, then condition (3) for ⟨t, g⟩ yields g(M ) ∈ W .
Since ⟨t, g⟩ ∈ Q certainly g(M ) ∈ Q. So h(M ) = g(M ) ∈ Q ∩ W = M ∗ . By
the proof of Lemma 2.21, the only other option is that M ∗ is the lowest node
of an added interval FW (in the terminology of the lemma). This lowest node is
equal to Q ∩ W , where W is a transitive node of t above M . Then by Claim 2.10
and condition (3) for ⟨t, g⟩, g(M ) ∈ W . g(M ) belongs to Q since ⟨t, g⟩ ∈ Q. So
h(M ) = g(M ) ∈ Q ∩ W = M ∗ .
For condition (4), it is enough to prove that for small N ∈ r, gN belongs to N ,
and if N ̸⊆ Q, then f N belongs to N too. This allows constructing hN inside
N . (The full function h can be constructed inside K from f and g.) Consider first
the case that N ∈ Q. Then N ⊆ Q because N is small, and N ∈ t by part (2) of
Corollary 2.31. By condition (4) for ⟨t, g⟩, gN ∈ N as required.
Suppose next that N ̸∈ Q. We prove that f N and gN both belong to N . The
proof in both cases is by induction on N .
If N is above Q, then N ∈ s and this immediately implies f N ∈ N . If there
are no nodes of transitive type between Q and N then Q ⊆ N and since ⟨t, g⟩ ∈ Q
it follows that gN = g ∈ N . If there are transitive nodes between Q and N , let
R be such a node. By induction gN ∩ R ∈ N ∩ R ⊆ N . Since R is above Q and
g ∈ Q, gN ∩ R = gN , so gN ∈ N .
A similar argument applies if N belongs to a residue gap of s in Q, say [Q∩W, W ),
using the fact that N is above Q ∩ W , and gN = gQ ∩ W in case Q ∩ W ⊆ N .
14
ITAY NEEMAN
The only remaining alternative is that Q is of small type and N belongs to an
added interval FW in the terminology of Lemma 2.21. In this case N contains
the smallest node of the interval, Q ∩ W , and gN = gQ ∩ W . Since g ⊆ Q,
gQ ∩ W = gW , which belongs to W by condition (4) for ⟨t, g⟩ and to Q since
g, W ∈ Q. So gN = gQ ∩ W ∈ Q ∩ W ⊆ N . N itself is equal to N ∗ ∩ W for some
N ∗ ∈ s. By condition (4) for ⟨s, f ⟩, f N ∗ ∈ N ∗ . Since N = N ∗ ∩ W , f N is an
initial segment of f N ∗ , hence it too belongs to N ∗ . It belongs to W by closure of
W under sequences of length < κ. So f N ∈ N ∗ ∩ W = N .
Remark 2.39. As usual, the part of the proof of Claim 2.38 handling condition
(4) of Definition 2.35 is not necessary in case κ = ω.
Corollary 2.40. Let M ∈ S ∪ T , and let ⟨t, g⟩ ∈ Pdec be an element of M . Then
there is ⟨r, h⟩ ∈ Pdec extending ⟨t, g⟩, with M ∈ r.
Proof. The proof of this, from Claim 2.38, is similar to the proof of Corollary 2.32
from Corollary 2.31.
3. Strong properness
This section includes some basic results about strong properness. The notion
and the results presented are due to Mitchell [6], except for Claim 3.8 which is due
to Friedman [3]. We also include some well known results about properness and
preservation of cardinals.
A condition p in a poset Q is a strong master condition for a model M if it forces
that Ġ ∩ M is generic for Q ∩ M . In other words it forces the generic filter to meet
every dense subset of Q ∩ M . The poset is strongly proper for M if every condition
in M can be extended to a strong master condition for M .
Recall that p is an ordinary master condition for M if it forces the generic object
to meet, inside M , every dense set of Q that belongs to M . Q is proper for M if
every condition in M can be extended to a master condition for M .
Remark 3.1. If p is a strong master condition for M , Q ∈ M , and M is sufficiently
elementary in a transitive model, then p is also an ordinary master condition for M .
To see this note that for any dense set D of Q that belongs to M , by elementarity
D ∩ M is dense in Q ∩ M .
Remark 3.2. Suppose Q ⊆ K is strongly proper for M ⊆ K. Let θ∗ be large
enough that K ⊆ H(θ∗ ) and suppose M ∗ ≺ H(θ∗ ) is such that M ∗ ∩ K = M . Then
Q is also strongly proper for M ∗ . This is immediate from the definitions, as only
M ∗ ∩ Q = M ∩ Q is relevant for determining strong properness.
The following claim gives standard consequences of properness. By the observation above, they are also consequences of strong properness.
Claim
Q over
(1)
(2)
3.3. Suppose M is elementary in H(θ∗ ) and Q ∈ M . Let G be generic for
V , and suppose that G includes a master condition for M . Then:
M [G] ≺ H(θ∗ )[G] and M [G] ∩ V = M .
Let f˙ ∈ M and suppose that f˙[G] is a function with ordinal domain. Let
τ = f˙ ∩ M . Then τ [G] = f˙[G]M .
Proof. The first part is well known. For the second, it is clear that τ [G] ⊆ f˙[G],
and (using the first part) that dom(τ [G]) ⊆ M . For the reverse inclusion, if α ∈ M
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
15
and f˙[G](α) = b, then there is some pair ⟨σ, q⟩ ∈ f˙ so that σ[G] = ⟨α, b′ ⟩ and
q ∈ G, and moreover for every such pair, b′ = b. Using the first condition, such a
pair ⟨σ, q⟩ can be found in M , so that it belongs to τ .
Call a poset proper for S if it is proper for every M ∈ S, and similarly with
strong properness.
Recall that X ⊆ P(Y ) is stationary in Y if it meets every club subset of P(Y ),
or more precisely, if for every function f : Y <ω → Y , there is u ∈ X which is closed
under f .
Claim 3.4. Suppose Q ∈ H(θ∗ ) is proper for S ∗ . Let δ < θ∗ be a cardinal and
suppose that for each α < δ, the set {M ∈ S ∗ | α ⊆ M and |M | < δ} is stationary
in H(θ∗ ). Then forcing with Q does not collapse δ.
Proof. This is a standard application of Claim 3.3. Let f˙ be a name for a function
into δ, with domain α < δ. Let p ∈ Q. Using the stationarity assumed in the claim,
find M ∈ S ∗ of size < δ with α ∪ {f˙, p, Q} ⊆ M ≺ H(θ∗ ). Let q ≤ p be a master
condition for M . By Claim 3.3, q forces the range of f˙ to be contained in M . Since
M ∈ V has size < δ, M ̸⊇ δ, and hence q forces f˙ to not be onto δ.
Claim 3.5. Suppose Q ⊆ K is strongly proper for S. Let δ be a cardinal and
suppose that for each α < δ, the set {M ∈ S | α ⊆ M and |M | < δ} is stationary
in K. Then forcing with Q does not collapse δ.
Proof. Let θ∗ be large enough that K, Q ∈ H(θ∗ ). Let S ∗ = {M ∗ ⊆ H(θ∗ ) |
M ∗ ∩ K ∈ S}. Then the stationarity assumed in the claim implies that {M ∈
S ∗ | α ⊆ M and |M | < δ} is stationary in H(θ∗ ). Moreover Q is strongly proper
for each M ∗ ∈ S ∗ by Remark 3.2, and hence proper for M ∗ by Remark 3.1. The
current claim now follows from Claim 3.4.
Lemma 3.6. Let G be generic for Q over V , let α be an ordinal, and let f =
f˙[G] ∈ V [G] be a function from α into the ordinals. Suppose that f˙, Q ∈ M , M
is elementary in some H(θ∗ ), and G includes a strong master condition for M . If
f M belongs to V , then the entire function f must belong to V .
Proof. Redefining the name f˙ if necessary, we may assume that all elements of f˙
ˇ = µ̌. Let τ = f˙ ∩ M .
are of the form ⟨⟨ξ, µ⟩ˇ, t⟩ where ξ < α, µ ∈ Ord, and t f˙(ξ)
˙
By Claim 3.3 and Remark 3.1, τ [G] = f [G]M . Suppose that τ [G] belongs to V .
We prove that f belongs to V .
We are assuming that G includes a strong master condition for M , and hence
G ∩ M is generic for Q ∩ M over V . Since τ ⊆ M and all elements of τ are of the
form ⟨⟨ξ, µ⟩ˇ, t⟩, τ is a Q ∩ M -name. τ [G] (with τ viewed as a Q name) is equal to
τ [G ∩ M ] (with τ viewed as a Q ∩ M name). Since τ [G ∩ M ] = τ [G] belongs to V ,
there is a condition r ∈ G ∩ M which forces (in Q ∩ M ) a specific value for τ . In
particular, for every ξ ∈ α ∩ M , there is µ ∈ Ord ∩ M such that every s ∈ Q ∩ M
with s ≤ r extends to t ∈ Q ∩ M so that ⟨⟨ξ, µ⟩ˇ, t⟩ ∈ f˙. By elementarity of M , it
follows that for every ξ ∈ α, there is µ ∈ Ord such that every s ∈ Q with s ≤ r
extends to t ∈ Q so that ⟨⟨ξ, µ⟩ˇ, t⟩ ∈ f˙. This implies that r in fact completely
forces, in Q, all values of f˙. So f = f˙[G] ∈ V .
Lemma 3.7. Suppose Q ⊆ K is strongly proper for S, let δ be a regular cardinal,
and suppose that {M ∈ S | |M | < δ} is stationary in K. Then forcing with Q does
16
ITAY NEEMAN
not add branches of length δ to trees in V . Precisely, suppose T is a tree in V , G is
generic for Q over V , and b ∈ V [G] is a branch through T of length δ. Then b ∈ V .
Proof. Without loss of generality we may assume that nodes of T are ordinals, so
that branches of length δ through T are functions from δ into ordinals. Suppose
for contradiction that b = ḃ[G] is a branch through T , of length δ, that belongs to
V [G] but not to V . Let r ∈ G be a condition forcing this.
Let θ∗ be large enough that K, Q, ḃ ∈ H(θ∗ ). Using the stationarity assumed
in the lemma, fix M ∗ ≺ H(θ∗ ) so that |M ∗ | < δ, r, ḃ, and Q belong to M ∗ , and
M ∗ ∩K ∈ S. Let p ≤ r be a strong master condition for M ∗ ∩K (equivalently, since
Q ⊆ K, a strong master condition for M ∗ ). Replacing the generic G if needed, we
may assume that p ∈ G.
Then since b ̸∈ V it follows by Lemma 3.6 that bM ∗ ̸∈ V . Let γ = sup(δ ∩ M ∗ ).
Since δ is regular and |M ∗ | < δ, γ is smaller than δ. Since T is a tree that belongs
to V , this implies that bγ belongs to V (it is the function that enumerates all T predecessors of the node b(γ) according to their order in T ). Since bM ∗ = bγM ∗
it follows that bM ∗ belongs to V , contradiction.
The next claim, which deals with the product of proper and strongly proper
posets, is an abstraction of Lemma 3 in Friedman [3].
Claim 3.8. Let A, P ∈ M ≺ H(θ∗ ). Suppose that A is strongly proper for M , and
P is proper for M . Then:
(1) If a and p are strong and ordinary master conditions for M in A and P
respectively, then ⟨a, p⟩ is a master condition for M in A × P.
(2) A × P is proper for M .
(3) If G is generic for A over V with a strong master condition for M , then in
V [G], P is proper for M [G].
Proof. We prove the first part of the claim. The other two parts are immediate
consequences of the first. Let D ∈ M be a dense subset of A × P, and let a and
p be strong and ordinary master conditions for M in A and P respectively. It is
enough to prove, for every such pair ⟨a, p⟩, that ⟨a, p⟩ is compatible with an element
of D ∩ M . Let Z = {b ∈ A ∩ M | (∃q ∈ M )⟨b, q⟩ ∈ D and q is compatible with
p}. The density of D, elementarity of M , and the fact that p is a master condition
for M in P, imply that Z is dense in A ∩ M . Since a is a strong master condition
for M there is b ∈ Z which is compatible with a. By definition of Z there is then
q ∈ M so that ⟨b, q⟩ ∈ D and q is compatible with p.
4. Sequence poset and strong properness
Let S and T be appropriate for κ, λ, and K. Let P be the sequence poset
associated to κ, S, T , and K.
Claim 4.1.
(1) Let s ∈ P and let Q be a node in s. Then s is a strong master
condition for Q.
(2) P is strongly proper for S ∪ T .
(3) If W ∈ S ∪ T , then P ∩ W is strongly proper for (S ∪ T ) ∩ W . For any
condition s ∈ P ∩ W and any node Q ∈ s, s is a strong master condition
for Q in P ∩ W .
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
17
Proof. Consider first condition (1). Suppose for contradiction that s is not a strong
master condition for Q. Extending s, we may fix a dense subset D of P ∩ Q and
assume that s forces the generic filter for P to avoid D. resQ (s) is a condition of
P that belongs to Q. By density of D, there is t ∈ D extending resQ (s). t belongs
to Q as D ⊆ Q. By Corollary 2.31, s and t are directly compatible. Let r witness
this. Then r is an extension of s that forces t into the generic object, contradicting
the fact that s forces the generic object to avoid D.
By Corollary 2.32, every condition u ∈ Q extends to a condition s with Q ∈ s,
which by (1) is a strong master condition for P. This establishes condition (2).
For condition (3), note that if s and t in the proof of condition (1) both belong to
W , then so does r, since by elementarity W is closed under intersections. Similarly
in the proof of condition (2), if u and Q belong to W then so does s. The same
proofs can therefore be used to get the strong properness of P ∩ W .
Claim 4.2. Let Pdec be the decorated sequence poset associated to κ, λ, S, and T .
Then Pdec is strongly proper for S ∪ T . Moreover, any condition ⟨t, g⟩ ∈ Pdec that
belongs to a node Q ∈ S ∪ T extends to a condition ⟨s, f ⟩ ∈ Pdec with Q ∈ s, and
any such condition ⟨s, f ⟩ is a strong master condition for Q.
Proof. Similar to the proof of Claim 4.1, but using Claim 2.38 and Corollary 2.40
instead of Corollaries 2.31 and 2.32.
Let W be a node. {W } is then a strong master condition for W , meaning that if
G is generic for P over V with {W } ∈ G, then Ḡ = G ∩ W is generic for P̄ = P ∩ W
over V . This implies that the forcing to add G can be broken into two stages: first
force with P̄ to add Ḡ, then force with a factor poset to add G over V [Ḡ]. It is
easy to check that the factor poset is simply the restriction of P to conditions s so
that W ∈ s and resW (s) ∈ Ḡ. (This poset belongs to V [Ḡ].) We continue with a
few claims and remarks on strong properness for the factor poset.
Claim 4.3. Let W be a transitive node, let P̄ = P ∩∪W , and let Ḡ be generic for P̄
over V . Let Ŝ = {M ∈ S | W ∈ M and M ∩ W ∈ Ḡ}. Suppose that, in V , S is
stationary in K. Then, in V [Ḡ], Ŝ is stationary in K.
Proof. Suppose for contradiction that f = f˙[Ḡ] : K <ω → K and no element of Ŝ is
closed under f . Let s̄ ∈ Ḡ force this.
Let θ∗ be large enough that K ∈ H(θ∗ ), and using the stationarity of S, find
∗
M ≺ H(θ∗ ) with P, K, W, s̄, f˙ ∈ M ∗ , so that M ∗ ∩ K ∈ S, meaning that M ∗ ∩ K
is a small node. Let M denote M ∗ ∩ K. Since W ∈ M , M ∩ W too is a small
node, and belongs to W . Let r̄ ∈ P̄ be an extension of s̄ with M ∩ W ∈ r̄. Such an
extension exists in P by Corollary 2.32, and by elementarity of W it can be taken
to belong to W , hence to P̄. Changing Ḡ if needed, we may assume that r̄ ∈ Ḡ. By
Claim 4.1, r̄ is a strong master condition for M ∩ W = M ∗ ∩ W in P̄ ⊆ W , hence
by Remark 3.2 also a strong master condition for M ∗ . By Remark 3.1 and Claim
3.3, M ∗ [Ḡ] ≺ H(θ∗ )[Ḡ] and M ∗ [Ḡ] ∩ V = M ∗ . Since f˙ ∈ M ∗ it follows that M ∗ is
closed under f , and hence so is M . This is a contradiction since r̄ forces M into
Ŝ.
Claim 4.4. Let W be a transitive node, let P̄ = P ∩ W , let Ḡ be generic for P̄ over
V , and let Q be the factor poset for adding a V generic for P extending Ḡ, over
V [Ḡ]. Let Ŝ be defined as in Claim 4.3. Then Q is strongly proper for Ŝ.
18
ITAY NEEMAN
Note that M [Ḡ] ∩ V = M for any M ∈ Ŝ, since {M ∩ W } is a strong master
condition for M in P ∩ W . Since Q ⊆ V , and since the definition of a strong master
condition for M ∗ in Q depends only on M ∗ ∩ Q, it follows that for M ∈ Ŝ, every
strong master condition for M in Q is also a strong master condition for M [Ḡ].
Claim 4.4 therefore shows that Q is strongly proper for {M [Ḡ] | M ∈ Ŝ}.
∪
Proof of Claim 4.4. Fix M ∈ S with W ∈ M and M ∩ W ∈ Ḡ. We have to show
that the factor poset Q is strongly proper for M . We show this by proving that:
(1) any condition s ∈ Q with M ∈ s is a strong master condition for M in Q; and
(2) any condition t ∈ Q that belongs to M can be extended to such an s.
Consider (1) first. Suppose for contradiction that s is not a strong master condition for M . Extending s if necessary we may assume it forces that the generic
avoids a specific D ⊆ Q ∩ M which is dense in Q ∩ M . So no extension r of s in Q
extends an element of D.
Recall that the factor poset Q consists of conditions u ∈ P so that W ∈ u and
resW (u) ∈ Ḡ. The fact that s is such a condition, M ∈ s, and W ∈ M , implies
that so is resM (s). (To see that resW (resM (s)) belongs to Ḡ, note that it is weaker
than resW (s), which belongs to Ḡ.) By density of D, there is some t ∈ D extending
resM (s). By Corollary 2.31 and since D ⊆ M , s and t are directly compatible in P.
To derive a contradiction we need to show they are compatible in the factor poset.
Let r witness that s and t are directly compatible in P. By Claim 2.33, resW (r)
is the closure of resW (s) ∪ resW (t) under intersections. From this, the fact that
resW (s) and resW (t) both belong to Ḡ, and the fact that Ḡ is a filter, it follows
that resW (r) too belongs to Ḡ. Hence r belongs to the factor poset, witnessing that
s and t are compatible in this poset. This completes the proof of (1).
(2) is a consequence of the proof of (1). Let t be a condition in the factor poset
that belongs to M . Let u be the condition {M ∩ W, W, M }. Then u is a condition
in the factor poset, resM (u) = {W } ⊆ t, and the proof of (1) shows that u and t
are compatible in the factor poset. Any condition s in the factor poset witnessing
this is an extension of t with M ∈ s.
Remark 4.5. We only proved strong properness of the factor poset for small nodes.
Similar strong properness for transitive nodes is also true. To be more precise, let
T̂ = {M ∈ T | W ∈ M }. It is easy to check that stationarity of T in V implies
stationarity of T̂ in V [Ḡ], and that the factor poset is strongly proper for T̂ . The
proofs are similar to the proofs of Claims 4.3 and 4.4, but simpler in several places.
Remark 4.6. We worked in Claims 4.3 and 4.4 under the assumption that W is
transitive. Similar claims hold for small W , and again the proofs are similar to the
proofs of the claims, but simpler. To be more precise, if W is small, than the factor
poset is strongly proper for Ŝ = {M ∈ S | W ∈ M } and for T̂ = {M ∈ T | W ∈
M }, and each of these sets is stationary in V [Ḡ] if it is stationary in V .
5. Initial applications
We can now give several quick applications of the two-type model sequence posets
defined in Section 2. We use the posets to obtain the tree property, to add clubs in
subsets of θ ≥ ω2 , to collapse to κ+ without adding branches of length κ+ to trees
in V , and to obtain a model of PFA with an inner model that is correct about ω2
but not about reals. The first two are the initial applications of the finite condition
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
19
posets of countable models developed by Friedman [2] and Mitchell [6]. Our poset
gives the same extensions, but the use of models of two types rather than only
countable models makes the proof conceptually simpler. The last is a sketch of the
argument of Friedman [3], but using the two-type model sequences.
5.1. The tree property (after Mitchell [6]). Let θ be a weakly compact cardinal. Let K = H(θ). Let T consist of all transitive W ≺ K of size < θ which are
countably closed, and let S consist of all countable M ≺ K. Both S and T are stationary. The fact that each W ∈ T is countably closed implies that M ∩ W ∈ W for
all M ∈ S. It is easy to check that if W ∈ T , M ∈ S, and W ∈ M , then M ∩W ≺K,
and therefore M ∩ W ∈ S. (By elementarity of M , and since W can be wellordered
inside K, there is a wellordering of W in M . This allows defining Skolem functions
for W , inside M . Using the Skolem functions and the elementarity of M one can
check that M ∩ W ≺ W , and this implies M ∩ W ≺ K.)
S and T are therefore appropriate for ω, ω1 , and K = H(θ). Let P = Pω,S,T ,K .
By Claim 4.1, P is strongly proper for every Q ∈ S ∪ T , and indeed every condition
s with Q ∈ s is a strong master condition for Q. It follows by Claim 3.5 that
forcing with P preserves ω1 and θ, using the stationarity of S for the former, and
the stationarity of T for the latter.
∪
Let G be generic for P over
∪ V . Then G is a set of nodes. Using Claim 2.10
it is clear that A = {W ∈ G | W is a transitive node} is increasing, both in ∈
and in ⊆. Moreover, by genericity and Corollary 2.32, A is unbounded in H(θ),
meaning that
∪ for every x ∈ H(θ), there is W ∈ A with x ∈ W . In particular this
means that A = H(θ).
∪
For each W ∈ A, let BW = {M ∈ G | M is a small node between W and W ∗ },
where W ∗ is the next element of A above W . Using Claim 2.10, it is clear that for
every W ∈ A, BW is increasing, both in ∈ and in ∪
⊆. By genericity and Corollary
2.32, for every x ∈ H(θ) there is a small node M in G with
∪ {W, x} ⊆∗M . Since all
∗
conditions
in
G
are
closed
under
intersection,
and
W
∈
G, M ∩ W too belongs
∪
to G, and must occur between W ∪and W ∗ since W ∈ M ∩ W ∗ . Letting x range
over elements of W ∗ it follows that BW = W ∗ .
Since BW is increasing, and all models in BW are countable, the length of BW
is at most ω1V . Again using the fact that all models in BW are countable, it follows
∪
V [G]
in V [G].
that W ∗ = BW has size at most ω1V = ω1
Since this is true for each W ∈ A, and since every ordinal below θ belongs to
some W ∈ A, it follows that in V [G], every ordinal between ω1 and θ is collapsed
V [G]
to ω1 . From this and the preservation of θ, it follows that ω2
= θ. We continue
to prove that in V [G], the tree property holds at θ.
Claim 5.1. In V [G], the tree property holds at θ = ω2 .
Proof. Suppose not, and let Ṫ ∈ V be a name for a tree witnessing this. We may
assume that elements of T = Ṫ [G] are pairs ⟨ξ, µ⟩ ∈ θ × ω1 , and that level ξ of T
consists exactly of {ξ} × ω1 . We may also assume that Ṫ ⊆ H(θ). Finally, we may
assume for definitiveness that it is forced outright in P = Pω,S,T ,H(θ) that there are
no cofinal branches through Ṫ .
By the Π11 indescribability of weakly compact cardinals (see Jech [5, Theorem
17.18]), there is an inaccessible cardinal κ < θ, so that:
(1) (H(κ); Ṫ ∩ H(κ)) ≺ (H(θ), Ṫ ).
20
ITAY NEEMAN
(2) It is forced in P̄ = P ∩ H(κ) that Ṫ ∩ H(κ) has no branches of length κ.
(3) It is forced in P̄ that κ is regular.
Let W = H(κ). W is then a node of transitive type, and changing G if necessary,
we may assume {W } ∈ G. By strong properness of P then, Ḡ = G ∩ W is generic
over V for P ∩ W . By the conditions above, in V [Ḡ], κ is a regular cardinal, and
T̄ = (Ṫ ∩ W )[Ḡ] is a tree on κ × ω1 , with no branches of length κ.
An application of Remark 3.1 and Claim 3.3 shows that (H(κ); T̄ ) ≺ (H(θ); T ),
and this implies that T̄ is equal to T κ. In particular any node on level κ of T
determines a branch of length κ in T̄ . Thus, in V [G], there are branches of length
κ through T̄ .
Let Q ∈ V [Ḡ] be the factor poset to add G, forcing over V [Ḡ]. Let Ŝ be as in
Claim 4.3. By the claim, Ŝ is a stationary set of countable elementary substructures
of H(θ) in V [Ḡ]. By Claim 4.4, Q is strongly proper for Ŝ. It follows by Lemma
3.7 that forcing with Q over V [Ḡ] does not add any branches of length κ to T̄ . But
this is a contradiction, since V [G] is an extension of V [Ḡ] by Q, and T̄ , which has
no branches of length κ in V [Ḡ], has such branches in V [G].
By similar proofs, but with κ > ω (and λ = κ+ ) one can of course obtain the tree
property at greater cardinals. The sequence poset can also be used to obtain the
tree property at two successive cardinals. For example, suppose θ is supercompact
and θ∗ > θ is weakly compact. Force with the poset P above to add G. Then follow
this by forcing, over V [G], with the sequence poset associated to ω1 , θ = ω2 , S
consisting of models of the form M [G] where θ ∈ M ≺ H(θ∗ ),
∪ M belongs to V , has
size < θ, has ordinal intersection with θ, and M ∩ H(θ) ∈ G, and T consisting
of models W [G] where W ≺ H(θ∗ ) is transitive and θ ∈ W . This will collapse all
ordinals between θ and θ∗ to θ, resulting in a model where θ∗ = ω3 , and where the
tree property holds at both θ and θ∗ .
However the proof in this case is substantially more involved, for a couple of
reasons. The second stage sequence poset forcing uses small nodes which are not
countably closed in V [G]. While θ and θ∗ are preserved by this forcing using strong
properness, preservation of ω1 requires a special argument. And while the proof of
the tree property at θ∗ is identical to the proof given above using strong properness
of the factor poset, the proof at θ is more involved, since factors of the second stage
poset to its small nodes are not strongly proper after forcing with the first stage.
5.2. Adding clubs with finite conditions (after Friedman [2] and Mitchell
[6]). Let θ ≥ ω2 be a regular cardinal. We show, under certain conditions which we
will explain below, how to add club subsets to stationary subsets of θ, using finite
conditions.
Lemma 5.2. Let K be a transitive set with K ∩ Ord = θ, satisfying enough of ZFC
for the properties listed at the beginning of Section 2. Let S and T be appropriate
for ω, ω1 , and K. Let Pdec be the decorated sequence poset associated to ω, S, T ,
and K.
(1) If S and T are stationary in K, then Pdec does not collapse ω1 and θ.
(2) If |K| = θ then Pdec is θ+ -c.c. and does not collapse cardinals above θ.
(3) If S ∪T is unbounded in K, meaning that for every x ∈ K there is N ∈ S ∪T
with x ∈ N , then Pdec adds a club subset of {sup(N ∩ Ord) | N ∈ S ∪ T }.
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
21
Proof. The first part is immediate from the strong properness of Pdec , given by
Claim 4.2, using Claim 3.5. The second part is clear as Pdec ⊆ K. For the
∪ third
part, let G be generic for Pdec over V , and let C = {sup(N ∩ Ord) | N ∈ G}. It
is clear that C ⊆ {sup(N ∩ Ord) | N ∈ S ∪ T }. We prove that C is club in θ.
Using Corollary 2.40, the fact that S ∪ T is unbounded in K implies that for
every α ∈ K, the set of conditions forcing an ordinal above α into C is dense. It
follows that C is unbounded in θ = K ∩ Ord.
It remains to show that C is closed. Let α < θ, and let ⟨s, f ⟩ ∈ Pdec force
that α ̸∈ C. It is enough to show that ⟨s, f ⟩ can be extended to a condition
forcing that α is not a limit point of C. Let Q be the first node in s so that
sup(Q ∩ Ord) ≥ α. (Extending s if necessary we may assume there is such a
node, using the unboundedness of C.) Let M be the largest node of s below Q.
Since ⟨s, f ⟩ forces α outside C, sup(Q ∩ Ord) > α. Hence there is ξ ≥ α which
belongs to Q. Let f ′ be the function on s that differs from f only on M , with
f ′ (M ) = f (M ) ∪ {ξ}. Then ⟨s, f ′ ⟩ ∈ Pdec and ⟨s, f ′ ⟩ ≤ ⟨s, f ⟩. Moreover, in every
extension ⟨t, g⟩ of ⟨s, f ′ ⟩, the successor of M is a node M ∗ with ξ ∈ M ∗ , and hence
sup(M ∗ ∩ Ord) > α. It follows that ⟨s, f ′ ⟩ forces that C has no elements between
sup(M ∩ Ord) and α, and in particular α is not a limit point of C.
By Lemma 5.2, adding a club subset of U ⊆ θ using finite conditions reduces
to finding appropriate S and T so that {sup(N ∩ Ord) | N ∈ S ∪ T } ⊆ U . To
ensure preservation of cardinals, S and T should be stationary in K, and K itself
should have size θ. We continue to describe situations, taken from Friedman [2]
and Mitchell [6], where this can be done.
Recall that U ⊆ θ is fat if for every club B ⊆ θ, U ∩ B contains a club of order
type ω1 + 1. For α < θ of uncountable cofinality, we say that U is locally fat at α
if U ∩ α contains a countably closed unbounded subset of α, or, more precisely, if
there is a club E ⊆ α so that U ∩ α ⊇ {ξ ∈ E | cof(ξ) = ω}.
Claim 5.3. Let U be a fat subset of θ, which is stationary on points of uncountable
cofinality. Then there is Ū ⊆ U , still stationary on points of uncountable cofinality,
so that for every α ∈ Ū of uncountable cofinality, Ū is locally fat at α. Moreover
Ū can be found so that all its elements have cofinality ω or ω1 .
Proof. Let X = {α ∈ U | cof(α) = ω1 and U ∩ α contains a club subset of α}. Let
B = θ − X. U ∩ B cannot contain a club of order type ω1 + 1, since the top point
of such a club would belong to X. Since U is fat, it follows that B cannot contain
a club, and therefore X is stationary. Set Ū = X ∪ {ξ ∈ U | cof(ξ) = ω}.
Recall that R is a thin stationary subset of P<ω1 (θ) if R is a set of countable
subsets of θ, R is stationary, and the set {δ < θ | {x ∩ δ | x ∈ R} has size δ} is
unbounded in θ. Equivalently this set is club in θ relative to ordinals of uncountable
cofinality.
Remark 5.4. If δ < θ → δ ω < θ then a thin stationary set on P<ω1 (θ) exists
trivially. Indeed P<ω (θ) itself is such a set. But thin stationary sets need not
always exist. For more on the existence and nonexistence of thin stationary sets
see Friedman–Krueger [4].
Suppose that there exists a thin stationary subset of P<ω1 (θ). Let R be such a
set. Let f : θ → P<ω1 (θ) be a function so that on a club Z of δ < θ of uncountable
22
ITAY NEEMAN
cofinality, {x ∩ δ | x ∈ R} ⊆ f ′′ δ. Such a function can be constructed using the
assumption that R is thin.
Let U ⊆ θ be stationary on points of uncountable cofinality, and locally fat at all
its elements of uncountable cofinality. Let c be a function so that for each α ∈ U
of uncountable cofinality, c(α) ⊆ α is a club witnessing that U is locally fat at α.
We show how to add a club subset of U , with finite conditions, without collapsing
ω1 , θ, or any cardinals above θ. This also shows how to add clubs in fat subsets of
θ which are stationary on points of uncountable cofinality, since any such set can
be thinned to U as above using Claim 5.3.
Let K = Lθ [f, Z, U, c] (constructing relative to Z, U , and the sets {⟨α, ξ⟩ | ξ ∈
f (α)} and {⟨α, ξ⟩ | ξ ∈ c(α)}). Note that |K| = θ. Let T consist of transitive W ∈
K which are elementary in (K; f, Z, U ), with sup(W ∩ Ord) ∈ U and cof(sup(W ∩
Ord)) uncountable. Let S consist of countable M ∈ K which are elementary in
(K; f, Z, U ), with sup(M ∩ Ord) ∈ U , and M ∩ W ∈ W for every W ∈ T which
belongs to M . An argument similar to that in the first paragraph of Subsection
5.1 shows that M ∩ W is elementary in (K; f, Z, U ) whenever M ∈ S, W ∈ T ,
and W ∈ M . Moreover, for such M and W , sup(M ∩ W ∩ Ord) ∈ U because U
is locally fat at sup(W ∩ Ord). (sup(W ∩ Ord) is a point of uncountable cofinality
in U . A club witnessing local fatness of U at sup(W ∩ Ord) belongs to M by
elementarity and inclusion of the function c in the structure K. This implies that
sup(M ∩ W ∩ Ord) belongs to this club, and hence sup(M ∩ W ∩ Ord) ∈ U .) Hence
M ∩ W ∈ S. It follows from this, using also the requirement M ∩ W ∈ W in the
definition of S, that S and T are appropriate for ω, ω1 , and K.
Claim 5.5. S and T are both stationary.
Proof. The stationarity of T is immediate from the stationarity of U on points of
uncountable cofinality and the fact that Lα [f, Z, U, c] ≺ K for a club of α < θ. We
prove that S is stationary. Let h be a function from K <ω into K. We have to prove
that there are countable M which belong to S and are closed under h.
Let W ∈ T be closed under h. Such W exists since T is stationary. Let Q ⊆ K
be countable, elementary in (K; f, Z, U ), and closed under h, with W ∈ Q and
Q∩Ord ∈ R. Such Q exists since R is stationary. Set M = Q∩W . Then M is closed
under h and elementary in (K; f, Z, U ). The local fatness of U at sup(W ∩ Ord),
and the inclusion of a club witnessing this in K, imply that sup(M ∩ Ord) =
sup(Q ∩ W ∩ Ord) ∈ U . Since Q ∩ Ord belongs to R and sup(W ∩ Ord) ∈ Z,
M ∩ Ord = (Q ∩ Ord) ∩ sup(W ∩ Ord) belongs to f ′′ sup(W ∩ Ord) ⊆ W . Since M
can be determined in K from M ∩ Ord it follows from this that M ∈ K and indeed
M ∈ W . Moreover for any W̄ ∈ T which belongs to M , a similar argument shows
that M ∩ W̄ = Q ∩ W̄ belongs to W̄ . This establishes that M ∈ S.
Now by Lemma 5.2, forcing with Pdec
ω,S,T ,K does not collapse ω1 , θ, or any cardinals above K, and adds a club subset to {sup(N ∩ Ord) | N ∈ S ∪ T }, which by
the definitions above is contained in U .
Remark 5.6. The requirement that R is thin can be weakened slightly, to require
that the sequence of sets {x ∩ δ | x ∈ R and δ ∈ x} is approachable on a stationary
subset of U of points of uncountable cofinality. Precisely this means that there is an
enumeration f of sets so that for stationarily many δ ∈ U of uncountable cofinality,
{x ∩ δ | x ∈ R and δ ∈ x} ⊆ f ′′ δ. (If we removed the part “δ ∈ x”, this would be
equivalent to thinness.) The proofs above go through essentially unmodified with
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
23
this condition, restricting T to W so that δ = sup(W ∩ Ord) is in the stationary
set witnessing approachability. The condition cannot be weakened further since
the existence of such R follows from the existence of stationary S and T which are
appropriate for K of size θ. To see this, let f enumerate K in order type θ. Then
the set {M ∩ W ∩ Ord | M ∈ S, W ∈ T , M and W are elementary with respect to
f , and W ∈ M } is stationary in P<ω1 (θ), and approachable on the stationary set
{sup(W ∩ Ord) | W ∈ T and W is elementary with respect to f }.
5.3. Collapsing without adding branches. Let κ be a cardinal and suppose
that κ<κ = κ. Let θ > κ+ be regular, with |H(θ)| = θ, and such that δ < θ →
δ κ < θ.
Set T to consist of all transitive W ≺ H(θ) so that |W | < θ and W is κ closed.
Set S to consist of all M ≺ H(θ) of size κ, with κ ∈ M , which are closed under
sequences of length < κ. Then S and T are appropriate for κ, κ+ , and H(θ). Our
assumptions on κ and θ imply that S and T are both stationary.
Let Sκ,θ be the poset Pκ,S,T ,H(θ) . By arguments similar to the collapsing arguments in Subsection 5.1, forcing with Sκ,θ collapses all cardinals between κ+ and θ
to κ+ . By Claim 2.8, Sκ,θ is <κ closed. It follows that forcing with Sκ,θ does not
collapse κ or any smaller cardinals, and indeed does not add sequences of ordinals
of length < κ. By Claim 4.1, Sκ,θ is strongly proper for S ∪ T . Since S and T
are both stationary, it follows using Remark 3.2, Remark 3.1, and Claim 3.3 that
forcing with Sκ,θ does not collapse κ+ and θ. Since Sκ,θ has size |H(θ)| = θ, it does
not collapse cardinals or change cofinalities above θ.
Altogether then, Sκ,θ has the same collapsing effects as Col(κ+ , <θ).
The collapse poset Col(κ+ , <θ) can be split into forcing first with an initial
segment Col(κ+ , γ) and then with a tail-end Col(κ+ , [γ, θ)). A similar splitting is
possible with Sκ,θ . For any W ∈ T , forcing with Sκ,θ below the condition {W } is,
by strong properness, the same as forcing first with Sκ,θ ∩ W , and then with the
factor poset Sκ,θ /(Sκ,θ ∩ W ). (Recall that the factor poset consists of all conditions
s ∈ Sκ,θ so that W ∈ s and resW (s) belongs to the generic added by Sκ,θ ∩ W .)
However, in contrast with Col(κ+ , <θ), Sκ,θ does not add branches of length κ+
or greater through trees in V . More precisely, if T is a tree in V , G is generic for
Sκ,θ over V , and f is a branch of T that belongs to V [G] and has length ≥ κ+ ,
then f ∈ V . This follows from strong properness for S and the stationarity of S,
by Lemma 3.7. Furthermore, the factor posets Sκ,θ /(Sκ,θ ∩ W ) too do not add
branches of length ≥ κ+ , to trees in the extension of V by Sκ,θ ∩ W . This again
follows by Lemma 3.7, this time using strong properness of the factor poset, given
by Claims 4.3 and 4.4.
(Note that the ordinary collapse poset, Col(κ+ , <θ), does add new branches of
length κ+ through trees in V . Indeed every segment Col(κ+ , γ) of the poset adds
such a branch, for example through the tree of functions into γ with domain < κ+ ,
ordered by extension.)
This property of Sκ,θ makes it a useful substitute for the ordinary collapse in
arguments that deal with the tree property.
One example involves models of the tree property at successors of singular cardinals together with failure of the Singular Cardinals Hypothesis. Neeman [8] produced such models for large singular cardinals. Sinapova [10] then produced such
models for ℵω2 . Passing from Neeman’s construction to Sinapova’s requires several
collapses, that ultimately will turn the large cardinal at the starting point to ℵω2 .
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ITAY NEEMAN
The initial impediment for combining such collapses with Neeman’s construction is
in adapting Lemma 3.2 of [8], which absorbs a branch through a tree from a generic
extension of a model to the model itself. Later impediments are of a similar nature.
Sinapova overcame these impediments in [10] using a very clever argument on
narrow systems, and systems of branches through them. If one were to replace
the ordinary collapses used in the construction, by the model sequence collapses
defined above, then less clever arguments would suffice. (The parallel of Lemma
3.2 in [8] for settings with the collapse is a consequence of the “no new branches”
property for factors of the model sequence collapses. Similar arguments, and other
arguments using closure of the model sequence collapses, can handle later issues in
the adaptation.)
5.4. A particular model of PFA (after Friedman [3]). Veličković [12, Theorem
3.13] proves that under the semi-proper forcing axiom (SPFA), every inner model
which is correct about ω2 , must contain all the reals. It is natural to ask if the same
result holds with PFA. Veličković–Caicedo [1] shows that a variant holds under PFA,
namely that if the inner model itself also satisfies PFA (and is correct about ω2 ),
then it must contain all reals. Indeed they prove this also for the bounded proper
forcing axiom, BPFA. However the original result fails under PFA. Friedman [3]
obtains a model of PFA with an inner model that is correct about ω2 , but does not
contain all reals (and similarly of BPFA, from smaller large cardinal assumptions).
His construction uses the side conditions of Friedman [2]. We briefly sketch an
adaptation of the construction to use the two-type model sequences.
Let θ be a supercompact cardinal. Let K = H(θ). Let S consist of all countable
elementary substructures of K. Let Z = {α < θ | H(α) ≺ H(θ) and H(α) is
countably closed}. Here α ranges over cardinals, and similarly in all contexts below
when we talk about H(α). Let T = {H(α) | α ∈ Z}. S and T are appropriate
for ω, ω1 , and K. Let P = Pω,S,T ,K . Let G be generic for P over V . As in
previous subsections, θ is turned to ω2 in the extension V [G], and apart from this
no cardinals are collapsed.
∪
We intend to work with the set {α | H(α) ∈ G}. As in previous subsections
the set is club in θ relative to Z, and this is all we need for the argument on PFA.
But in fact, in this case, the set is outright equal to Z. In other words, for every
α ∈ Z, H(α) ∈ G. This is immediate by a density argument using the following
claim:
Claim 5.7. Suppose that T consists exactly of nodes H(α) ≺ H(θ) so that α < θ
and H(α) is countably closed. Suppose also that all M ∈ S are elementary in H(θ).
Let s ∈ Pω,S,T ,H(θ) , and let H(α) ≺ H(θ) with α < θ be countably closed. Then
there is r ≤ s with H(α) ∈ r.
Moreover, one can arrange in addition that all new nodes in r, meaning nodes in
r − s, are either of transitive type or of the form N ∩ W where N ∈ s and W ∈ T .
Proof. Fix s and α. If s ⊆ H(α) then s ∪ {H(α)} is a condition. Suppose then that
s ̸⊆ H(α), and let M be the first node of s outside H(α). If M = H(α), take r = s.
If α ∈ M , then resM (s) ∪ {H(α)} is a condition that belongs to M and extends
resM (s). By Corollary 2.31 it is directly compatible with s, giving an extension r
of s which includes H(α). Since r is the closure of s ∪ {H(α)} under intersections,
the new nodes in r are H(α) itself, and intersections of small nodes of s with H(α).
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
25
Suppose finally that M is above H(α) and α ̸∈ M . In particular M is of small
type. Let α∗ be the first ordinal in M above α. Such an ordinal exists since
M ̸⊆ H(α). The minimality of α∗ implies that M ∩ H(α∗ ) = M ∩ H(α). From this,
the elementarity of M , and the elementarity of H(α), it follows that H(α∗ ) ≺ H(θ).
(Suppose not. Then there are a1 , . . . , an ∈ H(α∗ ) and a formula φ so that in H(θ)
there exists some y so that H(θ) |= φ(a1 , . . . , an , y), but there is no such y in H(α∗ ).
By elementarity of M , a1 , . . . , an can be found in M . They then belong to H(α).
By elementarity of H(α), y can be found in H(α), hence also in H(α∗ ).) It also
follows that α∗ has uncountable cofinality. So H(α∗ ) ∈ T . The argument of the
previous paragraph shows that there is an extension s∗ of s with H(α∗ ) ∈ s∗ . The
first node of s∗ above α has smaller von Neumann rank than the first node of s
above α, and by induction it follows that there is an extension r∗ of s∗ , hence also
of s, that contains H(α). In each of these two steps, the new nodes in the extension
are either of transitive type or of the form N ∩ W where N ∈ s and W ∈ T , using
the argument of the previous paragraph for the first step, and induction for the
second.
Remark 5.8. Let ⟨H(θ); A1 , . . . , Ak ⟩ be an expansion of H(θ) by finitely many
predicates. Then in Claim 5.7 and its proof, elementarity in H(θ) can be replaced
by elementarity in ⟨H(θ); A1 , . . . , Ak ⟩ throughout.
We now briefly sketch Friedman’s argument to force over V [G] to obtain PFA,
without collapsing any cardinals. The forcing will add reals, and so in the resulting
model of PFA, V [G] will be a submodel that is correct about ω2 and does not include
all reals.
We work throughout over the model V [G]. Our intention is to build a Laver
V [G]
over V [G], which forces PFA and does not collapse
iteration R of length θ = ω2
any cardinals. To reach PFA in the extension, call it V [G][I], we have to allow, at
stages α < θ of the iteration, posets which are proper in V [G ∩ H(α)][Iα]. One of
the key points in the argument is that properness transfers from V [G ∩ H(α)][Iα]
to V [G][Iα], allowing us to argue that the iteration is proper over V [G].
Let F ∈ V be a Laver function for the supercompact cardinal θ. Let θ∗ > θ and
let S ∗ be the set of countable M ≺ H(θ∗ ) with θ, F ∈ M . For each α ∈ Z ∪ {θ}, let
Sα∗ = {M [G ∩ H(α)] | M ∈ S ∗ , α ∈ M , and M ∩ H(α) ∈ G}. Let Qα be the factor
poset for adding G over V [G ∩ H(α)]. By Claim 4.4 and the comment following
the claim, Qα is strongly proper for Sα∗ .
A diagonal countable support iteration of length θ over V [G], leading to a final
poset R say, is the variant of a countable support iteration defined by placing the
following additional restriction on the sequences r that are allowed as conditions in
R and in its initial segments Rα: for every β that belongs to Z, rβ must belong to
V [G ∩ H(β)]. (In an ordinary iteration over V [G] one would have only rβ ∈ V [G].)
The notion is due to Friedman [3].
Let Z̄ ⊆ Z be the set of inaccessible α so that H(α) is closed under F . Working
in V [G], let R be the diagonal countable support length θ iteration of the posets
given by Rα-names F (α)[G ∩ H(α)] for α ∈ Z̄ so that, in Rα over V [G ∩ H(α)],
it is forced that F (α)[G ∩ H(α)] is proper with respect to (the Rα extensions of
models in) Sα∗ . (R is similar to the poset Q in Friedman [3].)
The use of diagonal iteration in the definition of R, together with the fact that
the individual posets being iterated all have size < θ, implies that R is θ-c.c. over
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V [G]. The proof of this involves a standard ∆ system argument; we note only that
the use of a diagonal iteration is essential for the argument as it limits the number
of conditions with any fixed support to less than θ. Since R is θ-c.c. over V [G], it
V [G]
does not collapse any cardinals ≥ ω2
= θ.
Recall that Sα∗ consists of structures of the form M [G ∩ H(α)]. By definition R
only uses F (α)[G ∩ H(α)] if it is forced to be proper for these structures. The next
claim shows that F (α)[G ∩ H(α)] is then forced to be proper also for the extended
structures M [G].
Claim 5.9. Let α be as above, and let M ≺ H(θ∗ ) be countable with α, F ∈ M .
Then it is forced, in Rα over V [G], that F (α)[G ∩ H(α)] is proper for M [G].
Proof. This follows by condition (3) in Claim 3.8, using the fact that the factor poset
Qα , which leads from V [G ∩ H(α)] to V [G], is strongly proper for M [G ∩ H(α)] and
remains so in the extension of V [G∩H(α)] by Rα. Claim 3.8 is used with the model
M [G ∩ H(α)][I] in V [G ∩ H(α)][I] where I is generic for Rα over V [G ∩ H(α)]. (In
passing from M [G∩H(α)] to M [G∩H(α)][I] we make implicit use of the properness
of Rα for M [G ∩ H(α)], which is proved below. Formally the two proofs are by
simultaneous induction.)
The poset R is, by the last claim, a countable support diagonal iteration of
proper posets in V [G], with properness restricted to elementary substructures in
Sθ∗ . Were it not for the use of a diagonal iteration, preservation of properness
under countable support iterations would directly imply that R is itself proper for
these substructures. With the use of a diagonal iteration, at each limit stage γ,
the inductive step of the preservation argument only yields that Rγ is proper for
structures M [G ∩ H(α∗ )] in V [G ∩ H(α∗ )], where α∗ is the first element of Z that is
≥ γ. Fortunately, by condition (3) of Claim 3.8 this implies properness of Rγ for
structures M [G] in V [G], allowing the inductive preservation argument to proceed,
and showing ultimately that R is proper in V [G] for structures in Sθ∗ . In particular
then R does not collapse ω1 over V [G].
Let I be generic for R over V [G]. We have seen above that V [G][I] and V [G]
have the same cardinals. It is clear that in V [G][I] there are more reals. Standard
arguments, directly from the definition of R, using the supercompactness of θ and
the fact that F is a Laver function, show that PFA holds in V [G][I].
6. The consistency of PFA using finite supports
Our main application of the sequence models poset is a new proof of the consistency of the Proper Forcing Axiom, that does not use preservation of properness
under countable support iterations. Instead, we will work with a finite support
iteration, and use side conditions from the sequence poset to enforce properness.
Let θ be a supercompact cardinal. Let F : θ → H(θ) be a Laver function. Set
K = H(θ). Let Z be the set of α < θ so that (H(α); F α) is elementary in
(H(θ); F ). For each α ∈ Z, let f (α) be the least cardinal so that F (α) ∈ H(f (α)).
Note that f (α) is smaller than the next element of Z above α. Set T to be the
set of models W = H(α) for α ∈ Z so that H(α) is countably closed (equivalently,
α has uncountable cofinality). Set S to be the set of countable models which are
elementary in (H(θ); F ). S and T are then stationary and appropriate for ω, ω1 ,
and K = H(θ).
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
27
We describe a poset A that forces PFA. Conditions in the poset have two components. One corresponds to a finite support iteration of proper posets given by
the Laver function F . This is similar to the standard consistency proof for PFA,
except that finite supports are used instead of countable. The other component
consists simply of conditions in the model sequence poset Pω,S,T ,H(θ) . We will connect the two components by restricting conditions in the first to master conditions
for models in the second.
Definition 6.1. Conditions in the poset A are pairs ⟨s, p⟩ so that:
(1) s is a condition in Pω,S,T ,H(θ) . In other words it is a finite, ∈-increasing
sequence of models from S ∪ T , closed under intersections.
(2) p is a partial function on θ, with domain contained in the (finite) set {α <
θ | H(α) ∈ s and A∩H(α) “F (α) is a proper poset”}.
(3) For all α ∈ dom(p), p(α) ∈ H(f (α)).
(4) For all α ∈ dom(p), A∩H(α) p(α) ∈ F (α).
(5) For all α ∈ dom(p) and each small node M ∈ s so that α ∈ M , ⟨s ∩
H(α), pα⟩ A∩H(α) “p(α) is a master condition for M ”.
The ordering on A is the following: ⟨s∗ , p∗ ⟩ ≤ ⟨s, p⟩ iff s∗ ≤ s and for every
α ∈ dom(p), ⟨s∗ ∩ H(α), p∗ α⟩ A∩H(α) p∗ (α) ≤F (α) p(α).
To avoid confusion in the arguments below, we mostly use the same pairings
of letters for conditions: we will use pairs ⟨s, p⟩, ⟨t, q⟩, ⟨u, h⟩ and ⟨a, h⟩, and will
mostly avoid other pairings such as ⟨s, q⟩ or ⟨t, p⟩.
Definition 6.1 is an induction on α ∈ Z ∪{θ}, as knowledge of A∩H(α) is needed
to evaluate the conditions on p(α). Conditions (2)–(4) are the standard conditions
in an iteration of proper posets given by the Laver function F . Condition (5)
connects this iteration with the side conditions given by Pω,S,T ,H(θ) .
Remark 6.2. If α ∈ Z, then A ∩ H(α) is definable in (H(θ); F ) from α. This is
because F α is definable, and so are S ∩H(α) and T ∩H(α). (The parts of S and T
below α can be defined using elementarity in (H(α); F α) instead of elementarity
in (H(θ); F ), and this can be done inside (H(θ); F ) with α as parameter.) In
particular it follows that A ∩ H(α) ∈ M for every M ∈ S with α ∈ M .
Remark 6.3. Condition (5) involves some abuse of notation, since M is not an
elementary substructure of the extension of H(θ) by A ∩ H(α). What we mean
precisely is that ⟨s ∩ H(α), pα⟩ A∩H(α) “p(α) is a master condition for M [Ġα ]”
where Ġα names the A ∩ H(α) generic.
Remark 6.4. Condition (5) holds for α and M iff it holds for α and M ∩ H(γ),
whenever γ ∈ Z ∪ {θ} is larger than α. The reason is that F (α) ∈ H(f (α)) ⊆ H(γ),
and so being a master condition for M in the interpretation of F (α) is equivalent
to being a master condition for M ∩ H(γ).
Claim 6.5. Condition (5) in Definition 6.1 is equivalent to the same condition
with the restriction to “M ∈ s so that α ∈ M ” replaced by restriction to “M ∈ s
which occur above H(α) in s and so that there are no transitive nodes in s between
H(α) and M ”.
Proof. It is clear that the original condition implies the version in the claim, since
by Claim 2.10, α ∈ M for every M which occurs above H(α) in s with no transitive
nodes between H(α) and M . For the converse, let W ∗ be the first transitive node of
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ITAY NEEMAN
s above H(α) if there is one, and H(θ) otherwise. By Remark 6.4, being a master
condition for M in F (α) is equivalent to being a master condition for M ∩ W ∗ .
From this, the closure of s under intersection with W ∗ , and the fact that if α ∈ M
then α ∈ M ∩ W ∗ and hence M ∩ W ∗ occurs above H(α), it follows that being
a master condition for all M ∈ s with α ∈ M is a consequence of being a master
condition for all M ∈ s between H(α) and W ∗ .
For β ∈ Z ∪ {θ}, let Aβ denote the poset given by Definition 6.1 with the added
restriction that dom(p) ⊆ β. For β ∈ Z this poset is related to A ∩ H(β), but
the two are not the same, since the latter restricts the side conditions to belong to
H(β), while the former does not. Aθ is equal to A.
Claim 6.6. Let α < β belong to Z ∪ {θ}. Let ⟨s, p⟩ ∈ Aβ with W = H(α) ∈ s. Let
⟨t, q⟩ ∈ A ∩ H(α) extend ⟨s ∩ H(α), pα⟩. Then ⟨s, p⟩ and ⟨t, q⟩ are compatible in
Aβ . Moreover this is witnessed by the condition ⟨u, h⟩ with u = t ∪ s, hα = q, and
h[α, β) = p[α, β).
Proof. s ∩ H(α) is equal to resW (s), so by Corollary 2.31, t and s are directly
compatible in Pω,S,T ,H(θ) . Let u witness this. u is the closure of s ∪ t under
intersections, and since t extends an initial segment of s, u is simply equal to s ∪ t.
Let h = q ∪ p[α, β). It is easy, with some uses of Claim 6.5, to check that ⟨u, h⟩
is a condition in Aβ and that it extends both ⟨t, q⟩ and ⟨s, p⟩.
Lemma 6.7. Let β belong to Z ∪ {θ}.
(1) Let ⟨s, p⟩ ∈ Aβ and let W = H(α) be a transitive node in s. Then ⟨s, p⟩ is
a strong master condition for W in Aβ .
(2) Let ⟨s, p⟩ ∈ Aβ , let W ∈ T , and suppose ⟨s, p⟩ ∈ W . Then ⟨s ∪ {W }, p⟩ is
a condition in Aβ . (It trivially extends ⟨s, p⟩.)
(3) Aβ is strongly proper for T .
Proof. Condition (2) is immediate from the definitions, using Corollary 2.32 which
implies that s ∪ {W } ∈ Pω,S,T ,H(θ) . Condition (3) is immediate from (1) and (2).
Condition (1) in case α < β follows from Claim 6.6: Let D be dense in Aβ ∩ H(α).
It is enough to prove that every ⟨s, p⟩ ∈ Aβ with H(α) ∈ s is compatible with a
condition in D. Fix ⟨s, p⟩. By density of D in Aβ ∩ H(α), there is ⟨t, q⟩ ∈ D which
extends ⟨s ∩ H(α), pα⟩. By Claim 6.6, ⟨s, p⟩ and ⟨t, q⟩ are compatible, and the
condition ⟨u, h⟩ given by the claim to witness this belongs to Aβ . In case α ≥ β,
condition (1) is proved similarly with a direct use of Corollary 2.31 instead of Claim
6.6.
Claim 6.8. Let Q be proper. Let κ be large enough that Q ∈ H(δ) for some
δ < κ. Let M0 ∈ M1 ∈ . . . Ml−1 be countable elementary substructures of H(κ)
with Q ∈ Mi for all i. Suppose that k < l and q ∈ Mk is a master condition for
M0 , . . . , Mk−1 . Then there is q ∗ ≤ q which is a master condition for M0 , . . . , Ml−1 .
Proof. By standard arguments using the fact that κ is larger than the least δ with
Q ∈ H(δ), for every countable M ≺ H(κ), and every r ∈ M , there is r∗ ≤ r which
is a master condition for M . The claim follows by successive applications of this
fact, setting rk = q ∈ Mk , obtaining for each i ≥ k a master condition ri+1 ≤ ri for
Mi in Mi+1 , and taking q ∗ = rl . ri+1 can be found in Mi+1 by elementarity.
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
29
Claim 6.9. Let ⟨s, p⟩ ∈ A. Suppose H(α) ∈ s, and α ̸∈ dom(p). Let M be a
small node of s and let ⟨t, q⟩ ∈ A ∩ M . Suppose that α ∈ dom(q) and ⟨s, p⟩ ≤
⟨t, qθ − {α}⟩. Suppose further that resM (s) − H(α) ⊆ t. Then there is p′ extending
p with dom(p′ ) = dom(p) ∪ {α} and so that ⟨s, p′ ⟩ is a condition in A extending
⟨t, q⟩.
Proof. Let W be the first transitive node of s above H(α) if there is one, and H(θ)
otherwise. Let N0 ∈ · · · ∈ Nl−1 list the small nodes of s above H(α) and below W .
Since q ∈ M , dom(q) is finite, and α ∈ dom(q), we have α ∈ M . This implies that
M ∩W appears on the list N0 , . . . , Nl−1 . Fix k so that M ∩W = Nk . Since ⟨t, q⟩ ∈ A
and α ∈ dom(q), A∩H(α) “F (α) is a proper poset and q(α) ∈ F (α)”. Moreover
for every small node N ∈ t with α ∈ N , it is forced by ⟨t ∩ H(α), qα⟩, and hence
also by ⟨s ∩ H(α), pα⟩, that q(α) is a master condition for N . In particular, since
t ⊇ resM (s) − H(α) ⊇ {N0 , . . . , Nk−1 }, this holds for N0 , . . . , Nk−1 .
By Claim 6.8 there is a name q̇ ∗ which is forced by ⟨s ∩ H(α), pα⟩ to be a
master condition for all models N0 , . . . , Nl−1 , and to extend q(α). q̇ ∗ can be picked
in H(f (α)).
Set p′ = p ∪ {α 7→ q̇ ∗ }. It is easy using Claim 6.5 to check that ⟨s, p′ ⟩ is a
condition. It extends ⟨t, q⟩ by the claim assumption that ⟨s, p⟩ ≤ ⟨t, qθ − {α}⟩. Claim 6.10. Let ⟨s, p⟩, ⟨t, q⟩ ∈ A. Let M be a small node of s and suppose that
⟨t, q⟩ ∈ M . Suppose that for some δ < θ, ⟨s, p⟩ extends ⟨t, qδ⟩ and dom(q) − δ
is disjoint from dom(p). Suppose further that resM (s) − H(δ) ⊆ t. Then there is
p′ extending p so that dom(p′ ) = dom(p) ∪ (dom(q) − δ) and so that ⟨s, p′ ⟩ is a
condition in A extending ⟨t, q⟩.
Proof. Immediate by successive applications of Claim 6.9, going over all α ≥ δ in
dom(q) in increasing order.
Corollary 6.11. Let M be a small node and let ⟨t, q⟩ ∈ A ∩ M . Then there is
⟨s, p⟩ ≤ ⟨t, q⟩ with M ∈ s.
Proof. By Lemma 2.21, {M } and t are directly compatible. Let s witness this.
Note that resM (s) = t by Corollary 2.31. Now apply Claim 6.10 with δ = 0 to
⟨s, ∅⟩ and ⟨t, q⟩.
Lemma 6.12. Let β ∈ Z ∪ {θ}. Let ⟨s, p⟩ be a condition in Aβ . Let θ∗ > θ and
let M ∗ ≺ H(θ∗ ) be countable with θ, F, β ∈ M ∗ . Let M = M ∗ ∩ H(θ) and suppose
that M ∈ s. Then:
(1) For every D ∈ M ∗ which is dense in Aβ , there is ⟨t, q⟩ ∈ D ∩ M ∗ which is
compatible with ⟨s, p⟩. Moreover there is ⟨s∗ , p∗ ⟩ ∈ Aβ extending both ⟨s, p⟩
and ⟨t, q⟩, so that resM (s∗ ) − H(β) ⊆ t, and all small nodes of s∗ above β
and outside M are either nodes of s or of the form N ′ ∩ W where N ′ is a
small node of s and W ∈ T .
(2) ⟨s, p⟩ is a master condition for M ∗ in Aβ .
Proof. Condition (2) for ⟨s, p⟩ is immediate from condition (1) for all extensions
of ⟨s, p⟩. We prove condition (1) by induction on β. If β is the first element of Z
then Aβ is isomorphic to Pω,S,T ,H(θ) , and the condition holds by Corollary 2.31.
We handle the limit and successor cases below.
30
ITAY NEEMAN
Suppose first that β is a limit point of Z ∪ {θ}. Let β̄ = sup(β ∩ M ∗ ). This may
be β itself if cof(β) = ω. Let δ < β̄ in Z ∩ M ∗ be large enough that dom(p) ∩ β̄ ⊆ δ.
Such δ exists since dom(p) is finite, while β̄ is a limit point of Z ∩ M .
Let E be the set of conditions ⟨t, q̄⟩ ∈ Aδ which extend to conditions ⟨t, q⟩ ∈ D
with qδ = q̄. E is dense in Aδ , and belongs to M ∗ . By induction there is ⟨t, q̄⟩ ∈
E ∩ M ∗ which is compatible with ⟨s, pδ⟩. Moreover there is ⟨s∗ , p1 ⟩ ∈ Aδ which
extends both ⟨t, q̄⟩ and ⟨s, pδ⟩ = ⟨s, pβ̄⟩, with resM (s∗ ) − H(δ) ⊆ t, and so that
all small nodes of s∗ above δ and outside M are either nodes of s or of the form
N ′ ∩ W where N ′ is a small node of s and W ∈ T .
Let ⟨t, q⟩ ∈ D witness that ⟨t, q̄⟩ ∈ E. Using elementarity of M ∗ , pick q inside
∗
M . By Claim 6.10, there is p2 extending p1 so that dom(p2 ) = dom(p1 )∪(dom(q)−
δ), and so that ⟨s∗ , p2 ⟩ extends ⟨t, q⟩.
Set p∗ = p2 ∪ p[β̄, β). It is enough now to prove that ⟨s∗ , p∗ ⟩ is a condition in
Aβ . (By construction it extends both ⟨t, q⟩ and ⟨s, p⟩.) Fix α ∈ dom(p)[β̄, β), and
fix a small node N ∈ s∗ with α ∈ N . We must check that p∗ (α) = p(α) is forced by
⟨s∗ ∩ H(α), p∗ α⟩ to be a master condition for N . It is enough to check that this is
forced by ⟨s ∩ H(α), pα⟩.
If N belongs to M then N ⊆ M , and since α ∈ [β̄, β) this contradicts the fact
that α ∈ N . So N must be outside M . It follows by properties of s∗ above that
N is either a node of s or an intersection N ′ ∩ W where N ′ is a small node in s
and W ∈ T . If N is a node of s then ⟨s ∩ H(α), pα⟩ forces p(α) to be a master
condition for N , because ⟨s, p⟩ is a condition in Aβ . The same is true in case N
has the form N ′ ∩ W for N ′ ∈ s and W ∈ T , using Remark 6.4. This completes
the proof of the limit case of the lemma.
Suppose next that β is a successor point of Z. Let α be the predecessor of β in Z.
By elementarity of M ∗ , α ∈ M ∗ . For expository simplicity, fix G which is generic
for Aα over V , with ⟨s, pα⟩ ∈ G. By induction ⟨s, pα⟩ is a master condition for
M ∗ in Aα , so M ∗ [G] ≺ H(θ∗ )[G] and M ∗ [G] ∩ V = M ∗ .
Suppose that H(α) is a node in s. (We will handle the case that H(α) ̸∈ s later.)
By Lemma 6.7, G ∩ H(α) is generic for A ∩ H(α) over V .
If it is not forced in A ∩ H(α) that F (α) is a proper poset, then Aβ is equal
to Aα and the lemma at β follows immediately by induction. Suppose then that
F (α) is forced to be a proper poset, and let Q = F (α)[G ∩ H(α)]. Q belongs to
M ∗ [G ∩ H(α)] ⊆ M ∗ [G].
Let W be the first transitive node of s above H(α) if there is one, and H(θ)
otherwise. Let N0 ∈ N1 ∈ . . . Nl−1 list the small nodes of s between H(α) and W ,
in increasing order. Let k < l be such that Nk = M ∩ W . Note that the nodes in
{N0 , . . . , Nl−1 } that belong to resM (s) are exactly the nodes N0 , . . . , Nk−1 .
Fix D ∈ M ∗ which is dense in Aβ . Let E be the set of u ∈ Q so that one of the
following two conditions holds:
(i) No extension of u is a master condition for the models Ni [G ∩ H(α)] for all
i < k.
(ii) There exists ⟨t, q⟩ ∈ D with ⟨t ∩ H(α), qα⟩ ∈ G, t ≤ resM (s), and q(α)[G ∩
H(α)] = u.
In the case of condition (ii), by Definition 6.1 and the fact that t ≤ resM (s), u is a
master condition for Ni [G ∩ H(α)] for all i < k.
All the parameters in the definition of E belong to M ∗ [G ∩ H(α)], and therefore
by elementarity so does E. The density of D in Aβ implies that E is dense in Q:
FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
31
˙ ∩ H(α)] be any condition in Q and suppose for contradiction it has
Let ū = ū[G
no extension in E. Let ⟨a, h⟩ ∈ G ∩ H(α) force this. Extending ū using failure of
(i) we may assume that it is a master condition for Ni [G ∩ H(α)] for all i < k.
We may assume ⟨a, h⟩ forces this by extending ⟨a, h⟩ if needed, and with further
extension we may assume also that resM (s) ∩ H(α) ⊆ a, since ⟨resM (s), ∅⟩ ∈ G.
Let a∗ = a ∪ resM (s) (which below W is exactly a ∪ {H(α), N0 , . . . , Nk−1 }). Let
˙ Then ⟨a∗ , h∗ ⟩ is a condition in Aβ , and any ⟨t, q⟩ ∈ D which
h∗ = h ∪ {α 7→ ū}.
extends it provides a contradiction to the fact that ū˙ is forced to have no extensions
in E.
We proceed by using the density of E, and properness of Q, to prove condition
(1) of the lemma.
If α ∈ dom(p), then by Definition 6.1, p(α)[G ∩ H(α)] is a master condition for
M ∗ [G ∩ H(α)] in Q. It follows using the density of E that there is u ∈ E ∩ M ∗ [G ∩
H(α)], and u∗ which extends both p(α)[G∩H(α)] and u. Since u∗ extends p(α)[G∩
H(α)], it is a master condition for Ni [G ∩ H(α)] for all i < l, and equivalently for
all small nodes N of s with α ∈ N .
If α ̸∈ dom(p), then u and u∗ with the same properties can be obtained as
follows: First let v ∈ Q be a master condition for Ni [G ∩ H(α)] for all i < l. This
is possible using Claim 6.8. Then obtain u and u∗ as in the previous paragraph,
starting from v instead of p(α)[G ∩ H(α)].
Since u∗ ≤ u is a master condition for each Ni [G ∩ H(α)], the membership of u
in E must hold through condition (ii) in the definition of E. Let ⟨t, q⟩ witness the
condition. Using the elementarity of M ∗ [G ∩ H(α)], pick ⟨t, q⟩ in this model. Since
M ∗ [G ∩ H(α)] ∩ V = M ∗ , ⟨t, q⟩ belongs to M ∗ .
⟨t, q⟩ is a condition in Aβ , and in particular q(α) is forced by ⟨t∩H(α), qα⟩ to be
a master condition for all small nodes N ∈ t with α ∈ N . Since q(α)[G ∩ H(α)] = u
and ⟨t ∩ H(α), qα⟩ ∈ G, u is a master condition for these nodes.
By Corollary 2.31, t and s are directly compatible. Let r witness this. By the
same corollary, resM (r) = t and hence the small nodes of r inside M are nodes of
t. Again by the corollary, the small nodes of r outside M are either nodes of s
or intersections of small nodes of s with transitive nodes of t. Since u is a master
condition for small N ∈ t with α ∈ N , and u∗ ≤ u is a master condition for small
N ∈ s with α ∈ N , it follows using Remark 6.4 that u∗ is a master condition for
all N ∈ r with α ∈ N .
Let u̇ and u̇∗ name u and u∗ respectively. Let ⟨a, h⟩ ∈ G∩H(α) be strong enough
to force all the properties of u, u∗ , and ⟨t, q⟩ proved above. Extending ⟨a, h⟩ we
may assume it is stronger than both ⟨s ∩ H(α), pα⟩ and ⟨t ∩ H(α), qα⟩.
By Claim 2.33, r ∩ H(α) = resH(α) (r) is the closure of resH(α) (s) ∪ resH(α) (t) =
(s ∪ t) ∩ H(α) under intersections. Since ⟨a, h⟩ extends both ⟨s ∩ H(α), pα⟩ and
⟨t∩H(α), qα⟩, and since a is closed under intersections, r∩H(α) ⊆ a. By Corollary
2.31, a and r are then directly compatible, and this is witnessed by a ∪ r.
Let s∗ = a ∪ r. Let p∗ = h ∪ {α 7→ u̇∗ }. It is easy now to check that ⟨s∗ , p∗ ⟩ is a
condition in Aβ , and extends both ⟨s, p⟩ and ⟨t, q⟩. This, together with properties
of r and a proved above, completes the proof of condition (1) in case that β is a
successor, α is the predecessor of β in Z, and H(α) ∈ s.
Suppose finally that β is a successor point of Z, α is the predecessor of β in Z,
and H(α) ̸∈ s. By Claim 5.7 and Remark 5.8, there is an extension s′ of s with
H(α) ∈ s′ , so that the only added nodes, meaning nodes in s′ − s, are transitive
32
ITAY NEEMAN
nodes and intersections of small nodes of s with transitive nodes. Then using
Remark 6.4, ⟨s′ , p⟩ is a condition. By the arguments above the lemma holds for
⟨s′ , p⟩, and this implies that it holds for ⟨s, p⟩.
Corollary 6.13. Forcing with A preserves ω1 and θ as cardinals. All cardinals
between ω1 and θ are collapsed to ω1 .
Proof. Preservation of θ is immediate from the strong properness given by Lemma
6.7 and stationarity of T . Preservation of ω1 is immediate from the properness
given by Lemma 6.12 for A = Aθ , Corollary 6.11, and stationarity of S. The
arguments in Subsection 5.1 applied to the sequence poset component of A show
that all cardinals between ω1 and θ are collapsed to ω1 .
Lemma 6.14. The extension of V by A satisfies the proper forcing axiom.
Proof. This follows from the inclusion of a Laver iteration of proper posets in A.
The argument is standard but uses the strong properness given by Lemma 6.7
rather than the automatic strong properness that holds for iterations. We give a
brief sketch.
Suppose the lemma fails and let ⟨a, h⟩ ∈ A force that Q̇ and Ḋξ , ξ < ω1 , provide
a counterexample. Pick Q̇ so that it is outright forced to be proper. Let γ be
large enough that Q̇ and Ḋξ belong to H(γ). Since F is a Laver function and θ is
supercompact, there is θ̄ < θ, γ̄ < θ, Ṗ ∈ H(γ̄), and Ėξ ∈ H(γ̄) for ξ < ω1 , so that:
(∗) (H(γ̄); F θ̄, Ṗ, Ėξ ) embeds elementarily into (H(γ); F, Q̇, Ḋξ ) via an embedding, π say, with critical point θ̄, and F (θ̄) = Ṗ.
(To see that a structure and embedding witnessing (∗) exist note that by standard
use of the Laver function there is σ : V → M with σ(F )(θ) = Q̇ and σH(γ) ∈ M .
Then in M there exists a structure, namely (H(γ); F, Q̇, Ḋξ ), and an embedding,
namely σH(γ), which satisfy (∗) relative to σ(H(γ); F, Q̇, Ḋξ ). Pulling back using
the elementarity of σ gives the required structure and embedding in V .)
θ̄ may be picked large enough that ⟨a, h⟩ ∈ A ∩ H(θ̄). Then by Lemma 6.7,
⟨a ∪ {H(θ̄)}, h⟩ is a condition in A, and a strong master condition for H(θ̄).
Let G be generic for A with ⟨a ∪ {H(θ̄)}, h⟩ ∈ G. By strong properness, G ∩ H(θ̄)
is generic for A∩H(θ̄) over V . π extends trivially to an embedding of H(γ̄)[G∩H(θ̄)]
into H(γ)[G].
It is easy to check, from the definition of A, genericity of G, and the fact that
F (θ̄) = Ṗ, that {p(θ̄)[G ∩ H(θ̄)] | ⟨s, p⟩ ∈ G and θ̄ ∈ dom(p)} is generic for Ṗ[G ∩
H(θ̄)] over V [G ∩ H(θ̄)]. The image of this set under the extended embedding π is
a filter on Q = Q̇[G] that meets each of the sets Dξ = Ḋξ [G].
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Department of Mathematics, University of California Los Angeles, Los Angeles, CA
90095-1555
E-mail address: [email protected]